Handbook of Number Theory Ii

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HANDBOOK OF NUMBER THEORY II

J. Sandor´ Babes¸-Bolyai University of Cluj Department of Mathematics and Computer Science Cluj-Napoca, Romania and

B. Crstici formerly the Technical University of Timis¸oara Timis¸oara Romania

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON A C.I.P. Catalogue record for this book is available from the Library of Congress.

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All Rights Reserved C 2004 Kluwer Academic Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microﬁlming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands. Contents

PREFACE 7

BASIC SYMBOLS 9

BASIC NOTATIONS 10

1 PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES 15 1.1 Introduction ...... 15 1.2 Some historical facts ...... 16 1.3 Even perfect numbers ...... 20 1.4 Odd perfect numbers ...... 23 1.5 Perfect, multiperfect and multiply perfect numbers ...... 32 1.6 Quasiperfect, almost perfect, and pseudoperfect numbers ...... 36 1.7 Superperfect and related numbers ...... 38 1.8 Pseudoperfect, weird and harmonic numbers ...... 42 1.9 Unitary, bi-unitary, inﬁnitary-perfect and related numbers ...... 45 1.10 Hyperperfect, exponentially perfect, integer-perfect and γ -perfect numbers ...... 50 1.11 Multiplicatively perfect numbers ...... 55 1.12 Practical numbers ...... 58 1.13 Amicable numbers ...... 60 1.14 Sociable numbers ...... 72

References 77

2 GENERALIZATIONS AND EXTENSIONS OF THE MOBIUS¨ FUNCTION 99 2.1 Introduction ...... 99

1 CONTENTS

2.2 Mobius¨ functions generated by arithmetical products (or convolutions) ...... 106 1Mobius¨ functions defined by Dirichlet products ...... 106 2 Unitary Mobius¨ functions ...... 110 3 Bi-unitary Mobius¨ function ...... 111 4Mobius¨ functions generated by regular convolutions .... 112 5 K -convolutions and Mobius¨ functions. B convolution .... 114 6 Exponential Mobius¨ functions ...... 117 7 l.c.m.-product (von Sterneck-Lehmer) ...... 119 8 Golomb-Guerin convolution and Mobius¨ function ...... 121 9 max-product (Lehmer-Buschman) ...... 122 10 Infinitary convolution and Mobius¨ function ...... 124 11 Mobius¨ function of generalized (Beurling) integers ..... 124 12 Lucas-Carlitz (l-c) product and Mobius¨ functions ...... 125 13 Matrix-generated convolution ...... 127 2.3 Mobius¨ function generalizations by other number theoretical considerations ...... 129 1 Apostol’s Mobius¨ function of order k ...... 129 2 Sastry’s Mobius¨ function ...... 130 3Mobius¨ functions of Hanumanthachari and Subrahmanyasastri ...... 132 4 Cohen’s Mobius¨ functions and totients ...... 134 5 Klee’s Mobius¨ function and totient ...... 135 6Mobius¨ functions of Subbarao and Harris; Tanaka; and Venkataraman and Sivaramakrishnan ...... 136 7Mobius¨ functions as coefficients of the cyclotomic polynomial ...... 138 2.4 Mobius¨ functions of posets and lattices ...... 139 1 Introduction, basic results ...... 139 2Factorable incidence functions, applications ...... 143 3Inversion theorems and applications ...... 145 4Mobius¨ functions on Eulerian posets ...... 146 5 Miscellaneous results ...... 148 2.5 Mobius¨ functions of arithmetical semigroups, free groups, finite groups, algebraic number fields, and trace monoids ...... 148 1Mobius¨ functions of arithmetical semigroups ...... 148 2 Fee abelian groups and Mobius¨ functions ...... 151 3Mobius¨ functions of finite groups ...... 154

2 CONTENTS

4Mobius¨ functions of algebraic number and function-ﬁelds ...... 159 5Trace monoids and Mobius¨ functions ...... 161

References 163

3 THE MANY FACETS OF EULER’S TOTIENT 179 3.1 Introduction ...... 179 1 The infinitude of primes ...... 180 2 Exact formulae for primes in terms of ϕ ...... 180 3 Infinite series and products involving ϕ,Pillai’s (Cesaro’s)` arithmetic functions ...... 181 4 Enumeration problems on congruences, directed graphs, magic squares ...... 183 5Fourier coefficients of even functions (mod n) ...... 184 6 Algebraic independence of arithmetic functions ...... 185 7 Algebraic and analytic application of totients ...... 186 8 ϕ-convergence of Schoenberg ...... 187 3.2 Congruence properties of Euler’s totient and related functions .... 188 1 Euler’s divisibility theorem ...... 188 2Carmichael’s function, maximal generalization of Fermat’s theorem ...... 189 3 Gauss’ divisibility theorem ...... 191 4 Minimal, normal, and average order of Carmichael’s function ...... 193 5Divisibility properties of iteration of ϕ ...... 195 6 Congruence properties of ϕ and related functions ...... 201 7 Euler’s totient in residue classes ...... 204 8 Prime totatives ...... 206 9 The dual of ϕ, noncototients ...... 208 10 Euler minimum function ...... 210 11 Lehmer’s conjecture, generalizations and extensions .... 212 3.3 Equations involving Euler’s and related totients ...... 216 1 Equations of type ϕ(x + k) = ϕ(x) ...... 216 2 ϕ(x + k) = 2ϕ(x + k) = ϕ(x) + ϕ(k) and related equations ...... 221 3 Equation ϕ(x) = k,Carmichael’s conjecture ...... 225 4 Equations involving ϕ and other arithmetic functions .... 230 5 The composition of ϕ and other arithmetic functions .... 234 6 Perfect totient numbers and related results ...... 240

3 CONTENTS

3.4 The totatives (or totitives) of a number ...... 242 1 Historical notes, congruences ...... 242 2 The distribution of totatives ...... 246 3 Adding totatives ...... 248 4 Adding units (mod n) ...... 249 5 Distribution of inverses (mod n) ...... 250 3.5 Cyclotomic polynomials ...... 251 1 Introduction, irreducibility results ...... 251 2Divisibility properties ...... 253 3 The coefficients of cyclotomic polynomials ...... 256 4 Miscellaneous results ...... 261 3.6 Matrices and determinants connected with ϕ ...... 263 1 Smith’s determinant ...... 263 2 Poset-theoretic generalizations ...... 266 3Factor-closed, gcd-closed, lcm-closed sets, and related determinants ...... 270 4 Inequalities ...... 273 3.7 Generalizations and extensions of Euler’s totient ...... 275 1Jordan, Jordan-Nagell, von Sterneck, Cohen-totients .... 275 2 Schemmel, Schemmel-Nagell, Lucas-totients ...... 276 3Ramanujan’s sum ...... 277 4 Klee’s totient ...... 278 5 Nagell’s, Adler’s, Stevens’, Kesava Menon’s totients .... 278 6 Unitary, semi-unitary, bi-unitary totients ...... 281 7 Alladi’s totient ...... 282 8Legendre’s totient ...... 283 9 Euler totients of meet semilattices and finite fields ..... 285 10 Nonunitary, infinitary, exponential-totients ...... 287 11 Thacker’s, Leudesdorf’s, Lehmer’s, Golubev’s totients. Square totient, core-reduced totient, M-void totient, additive totient ...... 289 12 Euler totients of arithmetical semigroups, finite groups, algebraic number fields, semigroups, finite commutative rings, finite Dedekind domains ...... 292

References 295

4 SPECIAL ARITHMETIC FUNCTIONS CONNECTED WITH THE DIVISORS, OR WITH THE DIGITS OF A NUMBER 329 4.1 Introduction ...... 329

4 CONTENTS

4.2 Special arithmetic functions connected with the divisors of a number ...... 330 1 Maximum and minimum exponents ...... 330 2 The product of exponents ...... 332 3 Arithmetic functions connected with the prime power factors ...... 334 4 Other functions; the derived sequence of a number ..... 336 5 The consecutive prime divisors of a number ...... 337 6 The consecutive divisors of an integer ...... 342 7 Functional limit theorems for the consecutive divisors . . . 343 8 Miscellaneous arithmetic functions connected with divisors ...... 345 9 Arithmetic functions of consecutive divisors ...... 349 10 Hooley’s function ...... 360 11 Extensions of the Erdos¨ conjecture (theorem) ...... 363 12 The divisors in residue classes and in intervals ...... 363 13 Divisor density and distribution (mod 1) on divisors .... 366 14 The fractal structure of divisors ...... 367 15 The divisor graphs ...... 369 4.3 Arithmetic functions associated to the digits of a number ...... 371 1 The average order of the sum-of-digits function ...... 371 2 Bounds on the sum-of-digits function ...... 376 3 The sum of digits of primes ...... 379 4Niven numbers ...... 381 5 Smith numbers ...... 383 6 Self numbers ...... 384 7 The sum-of-digits function in residue classes ...... 387 8 Thue-Morse and Rudin-Shapiro sequences ...... 390 9 q-additive and q-multiplicative functions ...... 401 10 Uniform - and well - distributions of αsq (n) ...... 410 11 The G-ary digital expansion of a number ...... 414 12 The sum-of-digits function for negative integer bases .... 417 13 The sum-of-digits function in algebraic number ﬁelds .... 418 14 The symmetric signed digital expansion ...... 421 15 Inﬁnite sums and products involving the sum-of-digits function ...... 423 16 Miscellaneous results on digital expansions ...... 427

References 433

5 CONTENTS

5 STIRLING, BELL, BERNOULLI, EULER AND EULERIAN NUMBERS 459 5.1 Introduction ...... 459 5.2 Stirling and Bell numbers ...... 459 1 Stirling numbers of both kinds, Lah numbers ...... 459 2 Identities for Stirling numbers ...... 464 3 Generalized Stirling numbers ...... 469 4 Congruences for Stirling and Bell numbers ...... 488 5 Diophantine results ...... 507 6 Inequalities and estimates ...... 508 5.3 Bernoulli and Euler numbers ...... 525 1 Deﬁnitions, basic properties of Bernoulli numbers and polynomials ...... 525 2 Identities ...... 534 3 Congruences for Bernoulli numbers and polynomials. Eulerian numbers and polynomials ...... 539 4 Estimates and inequalities ...... 574

References 585

Index 619

6 Preface

The aim of this book is to systematize and survey in an easily accessible manner the most important results from some parts of Number Theory, which are connected with many other fields of Mathematics or Science. Each chapter can be viewed as an encyclopedia of the considered field, with many facets and interconnections with vir- tually almost all major topics as Discrete mathematics, Combinatorial theory, Numer- ical analysis, Finite difference calculus, Probability theory; and such classical fields of mathematics as Algebra, Geometry, and Mathematical analysis. Some aspects of Chapter 1 and 3 on Perfect numbers and Euler’s totient, have been considered also in our former volume ”Handbook of Number Theory” (Kluwer Academic Publishers, 1995), in cooperation with the late Professor D. S. Mitrinovico´ fBelgrade University, as well as Professor B. Crstici, formerly of Timis¸oara Technical University. However, there were included mainly estimates and inequalities, which are indeed very useful, but many important relations (e.g. congruences) were left out, giving a panoramic view of many other parts of Number Theory. This volume aims also to complement these issues, and also to bring to the attention of the readers (specialists or not) the hidden beauty of many theories outside a given field of interest. This book focuses too, as the former volume, on some important arithmetic functions of Number Theory and Discrete mathematics, such as Euler’s totient ϕ(n) and its many generalizations; the sum of divisors function σ(n) with the many old and new issues on Perfect numbers; the Mobius¨ function, along with its generalizations and extensions, in connection with many applications; the arithmetic functions related to the divisors, consecutive divisors, or the digits of a number. The last chapter shows perhaps most strikingly the cross-fertilization of Number theory with Combi- natorics, Numerical mathematics, or Probability theory. The style of presentation of the material differs from that of our former volume, since we have opted here for a more flexible, conversational, survey-type method. Each chapter is concluded with a detailed and up-to-date list of References, while at the end of the book one can find an extensive Subject index.

7 PREFACE

We have used a wealth of literature, consisting of books, monographs, journals, separates, reviews from Mathematical Reviews and from Zentralblatt fur¨ Mathe- matik, etc. This volume was not possible to elaborate without the kind support of many people. The author is indebted to scientists all over the world, for providing him along the years reprints of their papers, books, letters, or personal communi- cations. Special thanks are due to Professors A. Adelberg, G. Andrews, T. Agoh, R. Askey, H. Alzer, J.-P. Allouche, K. Atanassov, E. Bach, A. Blass, W. Borho, P. B. Borwein, D. W. Boyd, D. Berend, R. G. Buschman, A. Balog, A. Baker, B. C. Berndt, R. de la Breteche,` B. C. Carlson, C. Cooper, G. L. Cohen, M. Deaconescu, R. Dus- saud, M. Drmota, J. Desarm´ enien,´ K. Dilcher, P. Erdos,¨ P. D. T. A. Elliott, M. Eie, S. Finch, K. Ford, J. B. Friedlander, J. Feher,´ A. A. Gioia, A. Grytczuk, K. Gyory,¨ J. Galambos, J. M. DeKoninck, P. J. Grabner, H. W. Gould, E.-U. Gekeler, P. Hagis, Jr., D. R. Heath-Brown, H. Harborth, P. Haukkanen, A. Hildebrand, A. Hoit, F. T. Howard, L. Habsieger, J. J. Holt, A. Ivic,´ H. Iwata, K.-H. Indlekofer, F. Halter-Koch, H.-J. Kanold, M. Kishore, I. Katai,´ P. A. Kemp, E. Kratzel,¨ T. Kim, G. O. H. Katona, P. Leroux, A. Laforgia, A. T. Lundell, F. Luca, D. H. Lehmer, A. Makowski, M. R. Murthy, V. K. Murthy, P. Moree, H. Maier, E. Manstavicius,ˇ N. S. Mendelsohn, J.-L. Nicolas, E. Neuman, W. G. Novak, H. Niederhausen, C. Pomerance, S.ˇ Porubsky,´ L. Panaitopol, J. E. Pecariˇ c,´ Zs. Pales,´ A. Peretti, H. J. J. te Riele, B. Rizzi, D. Redmond, N. Robbins, P. Ribenboim, I. Z. Ruzsa, H. N. Shapiro, M. V. Subbarao, A. Sark´ ozy,¨ A. Schinzel, R. Sivaramakrishnan, J. Suranyi,´ T. Salˇ at,´ J. O. Shallit, K. B. Stolarsky, B. E. Sagan, I. Sh. Slavutskii, F. Schipp, V. E. S. Szabo,´ L. Toth,´ G. Tenenbaum, R. F. Tichy, J. M. Thuswaldner, Gh. Toader, R. Tijdeman, N. M. Temme, H. Tsumura, R. Wiegandt, S. S. Wagstaff, Jr., Ch. Wall, B. Wegner, M. Wojtowicz.´ The author wishes to express his gratitude also to a number of organizations whom he received advice and support in the preparation of this material. These are the Mathematics Department of the Babes¸-Bolyai University, the Alfred Renyi´ Institute of Mathematics (Budapest), the Domus Hungarica Foundation of Hungary, and the Sapientia Foundation of Cluj, Romania. The gratefulness of the author is addressed to the staff of Kluwer Academic Publishers, especially to Mr. Marlies Vlot, Ms. Lynn Brandon and Ms. Liesbeth Mol for their support while typesetting the manuscript. The camera-ready manuscript for the present book was prepared by Mrs. Georgeta Bonda (Cluj) to whom the author expresses his gratitude.

The author

8 Basic Symbols

f (x) = O(g(x)) or For a range of x-values, there is f (x) g(x) a constant A such that the inequality | f (x)|≤Ag(x) holds over the range

f (x) g(x) g(x) f (x) (or g(x) = O( f (x)))

f (x) f (x) = o(g(x)) lim = 0(g(x) = 0forx large). x→∞ g(x) The same meaning is used when x →∞ is replaced by x → α for any ﬁxed α

f (x) f (x) ∼ g(x) lim = 1(g(x) = 0forx large). x→∞ g(x) The same meaning when x →∞is replaced with x → α

f (x) g(x) There are c1, c2 such that c1g(x) ≤ f (x) ≤ c2g(x) for sufﬁciently large x (g(x)>0)

f (x) = (g(x)) f (x) = O(g(x)) does not hold

9 Basic Notations

ϕ(n) Euler’s totient function Jk(n) Jordan’s totient Sk(n) Schemmel’s totient C(n, r) Ramanujan’s sum ϕ(x, n) Legendre totient ∗ ϕ (n), ϕ∞(n), ϕe(n) unitary, inﬁnitary, exponential totient σ(n) sum of divisors function d(n) number of divisors function ω(n), (n) number of distinct, resp. total number, of prime factors of n σk(n) sum of kth powers of divisors of n ∗ ∗∗ σ (n), σ (n), σ∞(n), unitary, bi-unitary, inﬁnitary, exponential, # σe(n), σ (n) nonunitary sum of divisors functions ψ(n) Dedekind’s arithmetical function P(n) greatest prime factor of n λ(n) Liouville’s function; or Carmichael’s function s(n) = σ(n) − n, sum of aliquot, resp. unitary s∗(n) = σ ∗(n) − n aliquot, divisors of n µ(n) Mobius¨ function µ∗(n), µ∗∗(n), µ(e)(n) unitary, bi-unitary, exponential Mobius¨ functions µk(n) Mobius¨ function of order k µA(n) Narkiewicz Mobius¨ function ϕ(G) Euler’s totient of a group G µG (n), µ(G) Mobius¨ function of an arithmetical semigroup G,resp. of a group G

10 BASIC NOTATIONS

µ(x, y), µ)k(P), µM (t) Mobius¨ functions of posets, resp. of a trace monoid M µK (a) Mobius¨ function of an algebraic number ﬁeld ∗ ∗∗ T (n), T (n), Te(n) unitary, bi-unitary, exponential analogs of the product of divisors of n n(x) cyclotomic polynomial Vϕ(n) valence function of ϕ (i.e. number of solutions of ϕ(x) = n) E(n) Euler minimum function; or Erdos’¨ function ∞ ζ(s) = n−s(Re s > 1) Riemann’s zeta function n=1 S(n) Smarandache function; or Schinzel-Szekeres function H(n) maximum exponent of n, or harmonic numbers h(n) minimum exponent of n (n) Hooley’s function ω(v) Buchstab function ρ(v) Dickman function ( ), ( ) H1 n H1 n DeKoninck-Ivic´ function H2(n) Tenenbaum’s function T (n) product of divisors of n;orsum of iterated totients; or tangent numbers F(n) = 22n + 1 Fermat numbers t(n) set of totatives of n sq (n), s(n) sum of digits of n in base q, resp. base 10 , tq,m tn Thue-Morse sequences a(n) = (−1)e(n) Rudin-Shapiro sequence (−1)z(n) Zeckendorf sequence DN (xn), DN (xn) discrepancy, resp. uniform discrepancy of the sequence (xn) d(A) (asymptotic) density of A EX, VX expactation, resp. variance of the random variable X

11 BASIC NOTATIONS

dim X Hausdorff dimension dim fX fractal dimension φ(u) normal distribution function , , ∗ Bn Bn Bn Bernoulli numbers B(n) Bell numbers S(n, k), s(n, k) Stirling numbers s∗(n, k) unsigned Stirling numbers of the ﬁrst kind Sr (n, k) r-Stirling numbers St (n, k), st (n, k), Stirling numbers associated to ST (n, k), sT (n, k) the sequence t,resp. matrix T S(n, k,λ|θ) Howard’s degenerate weighted Stirling numbers S(n, k|θ),s(n, k|θ) Carlitz’ degenerate Stirling numbers S(n, k,α),S(n, k; α, β, γ ) Dickson-Stirling numbers; resp. Hsu-Shiue-Stirling numbers d(n, k), b(n, k) associated Stirling numbers S[n, k], s[n, k] q-Stirling numbers s∗[n, k] signless q-Stirling numbers of the ﬁrst kind Sp,q [n, k], sp,q [n, k] p, q-Stirling numbers [n] = [n]q , [n]p,q q-analogue, resp. p, q-analogue of the integer n xa,b(n) general factorial numbers P(n, k) = k!S(n, k) number of preferential arrangements S(x, y), s(x, y) Stirling numbers of the real numbers x, y Bn, Bn conjugate, resp. universal-Bernoulli numbers Bn(z) Bernoulli numbers of higher order n Bχ generalized Bernoulli numbers (χ a character) , , ∗, En En En En Euler numbers En(q), En|k(q), Hk(u, q) q-Euler numbers Gn Genocchi numbers βk(q), βk,χ (q) q-Bernoulli, resp. generalized q-Bernoulli-numbers Hk,χ (u, q) generalized q-Euler numbers

12 BASIC NOTATIONS

q(a, p), q(a, m) Fermat and Euler quotients wp Wilson quotient βn(q) modified q-Bernoulli numbers ∗( ) Bn q p-adic q-Bernoulli numbers ζq (s), ζq (s, x) q-Riemann ζ-function, resp, q-Hurwitz ζ-function Lq (s,χ) q − L-series G(x, q) q − log −-function A(n, k), a(n, k) Eulerian numbers Bn,k(q), Am,n(q) q-Eulerian numbers A(n, k,α) Dickson-Eulerian numbers A−(n, k) signed Eulerian numbers m ak,n generalized Eulerian numbers m E(m, k) = second order Eulerian numbers k a|b, a b a divides b, a does not divide b a ≡ b (mod n) n divides (a − b) for integers a, b a c ≡ (mod n) n|(ad − bc) for (b, n) = (d, n) = 1 b d n! = 1 · 2 ...n factorial of n !n = 0! + 1! + ···+(n − 1)! left-factorial of n (a, b) g.c.d. of a and b,oranordered pair [a, b] l.c.m. of a and b (x) Euler’s gamma function exp(z) ez [x]or[x]∗ integer part of x {x}= x − [x] fractional part of x p Legendre (or Jacobi) symbol q n binomial coefficient k n n = q-binomial coefficient k k q Fn, Ln Fibonacci, resp. Lucas - numbers νp(n) p-adic order of n [ai, j ]m×n m × n matrix of components aij f ∗ g Dirichlet convolution f g, f g unitary, resp. bi-unitary convolution

13 BASIC NOTATIONS

f ∗A g, f ◦ g regular (Narkiewicz) resp. K (Davison) -convolution (or composition of functions) f ∗ex g, f ∇g,(f ∗ g)∞ exponential, Golomb resp. inﬁnitary-convolution f ♦g, f #g max-product, resp. Cauchy product f ∗l−c g Lucas-Carlitz product f ∗G g matrix-generated convolution

14 Chapter 1

PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES

1.1 Introduction

The aim of this chapter is to survey the most important and interesting notions, results, extensions, generalizations related to perfect numbers. Many old, as well as new open problems will be stated, which will motivate - we do hope - many further research. This is one of the oldest subjects of Mathematics, with a considerable history. Some basic historical facts will be presented, as this will underline too our strong opinion on the role of perfect numbers in the development of Mathemat- ics. It is sufﬁcient to only mention here Fermat’s ”little theorem” of considerable importance in Number theory, Algebra, and more recently in Criptography. This theorem states that for all primes p and all positive integers a, p divides a p − a. Fermat discovered this result by studying perfect numbers, and trying to elaborate a theory of these numbers. One more example is the theory of primes in special sequences, and generally the classical theory of primes. Even perfect numbers involve the so called ”Mersenne primes”, of great importance in many parts of Number theory. Currently, about 39 such primes are known (39 as of 14-XI-2001, see e.g. http://www.stormloader.com/ajy/perfect/html), giving 39 known perfect numbers, all even. Recently (at the end of 2003) the 40th perfect number has been discovered. No odd perfect numbers are known, but we shall see on the part containing this theme, the most important and up-to-date results obtained along the centuries. An extension

15 CHAPTER 1 of perfect numbers are the ”amicable numbers” having a same old history, with considerable interest for many mathematicians. Many results, more generalizations, connections, analogies will be pointed out. Here the theory is filled again by a lot of unsolved problems. Along with the extensions of the notion of divisibility, there appeared many new notions of perfect numbers. These are e.g. the unitary perfect-, nonunitary perfect- , biunitary perfect-, exponential perfect-, infinitary perfect-, hyperperfect-, integer perfect-, etc., numbers. On the other hand, there appeared also the necessity of studying, by analogy with the classical case, such notions as: superperfect-, almost perfect-, quasiperfect-, pseudoperfect (or semi-perfect), multiplicatively perfect, etc., numbers. Some authors use different terminologies, so one aim is also to fix in the literature the exact terminologies and notations. Our aim is also to include results and references from papers published in certain little known journals (or unpublished results, obtained by personal communication to the authors).

1.2 Some historical facts

It is not exactly known when perfect numbers were first introduced, but it is quite possible that the Egypteans would have come across such numbers, given the way their methods of calculation worked (”unit fractions”, ”Egyptean fractions”). These numbers were studied by their mystical properties by Pythagoras, and his followers. For the Pythagorean school the ”parts” of a number are their divisors. A number which can be built up from their parts (i.e. summing their divisors) should be indeed wonderful, perfectly made by the God. God created the world in six days, and the number of days it takes the Moon to travel round the Earth is nothing else than 28. (For the number mysticism by Pythagoras’ school see U. Dudley [85]). These are the first two perfect numbers. The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times, and there is no record of these discoveries. The first recorded result concerning perfect numbers which is known occurs in Euclid’s ”Elements” (written around 300BC), namely in Proposition 36 of Book IX: ”If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.” Here ”double proportion” means that each number of the sequence is twice the preceding number. Since 1 + 2 + 4 +···+2k−1 = 2k − 1, the proposition states that: If, for some k > 1, 2k − 1isprime, then 2k−1(2k − 1) is perfect. (1)

16 PERFECT NUMBERS

Here we wish to mention another source for perfect numbers (usually overlooked by the Historians of mathematics) in ancient times, namely Plato’s Republic, where the so-called periodic perfect numbers were introduced. It is remarkable that 2000 years later when Euler proves the converse of (1) (published posthumously, see [97]) he makes no references to Euclid. However, Euler makes reference to Plato’s periodic perfect numbers. M. A. Popov [243] says that Euler’s proof was probably inspired by Plato. Another early reference seems to be at Euphorion (see J. L. Lightfoot [191]) a poet of the third century, B.C. The following significant study of perfect numbers was made by Nichomachus of Gerasa. Around 100 AD he wrote his famous ”Introductio Arithmetica” [227], which gives a classification of numbers into three classes: abundant numbers which have the property that the sum of their aliquot parts is greater than the number, deficient numbers which have the property that the sum of their aliquot parts is less than the number, and perfect numbers. Nicomachus used this classification also in moral terms, or biological analogies: ”... in the case of too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insuffi- ciencies...” ”... abundant numbers are like an animal with ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms...” ”... deficient numbers are like animals with a single eye,... one armed or one of his hands, less than five fingers, or if he does not have a tongue.” In the book by Nicomachus there appear five unproved results concerning perfect numbers: (a) the n-th perfect number has n digits; (b) all perfect numbers are even; (c) all perfect numbers end in 6 and 8 alternately; (d) every perfect number is of the form 2k−1(2k − 1), for some k > 1, with 2k − 1 = prime; (e) there are infinitely many perfect numbers. Despite the fact that Nicomachus offered no justification of his assertions, they were taken as fact for many years. The discovery of other perfect numbers disproved immediately assertions (a) and (c). On the other hand, assertions (b), (d), (e) remain unproved practically even in our days. The Arab mathematicians were also fascinated by perfect numbers and Thabit ibn Quarra wrote ”Treatise on amicable numbers” in which he examined when numbers of the form 2n p (p prime) can be perfect. He proved also ”Thabit’s rule” (see the section with amicable numbers). Ibn al-Haytham proved a partial converse to Euclid’s proposition (1), in the unpublished work ”Treatise on analysis and synthesis” (see [247]).

17 CHAPTER 1

Among the Arab mathematicians who take up the Greek investigation of perfect numbers with great enthusiasm was Ismail ibn Ibrahim ibn Fallus (1194-1239) who wrote a treatise based on Nicomachus’ above mentioned text. He gave also a table of ten numbers claiming to be perfect. The first seven are correct, and in fact these are indeed the first seven perfect numbers. For details of this work see the papers by S. Brentjes [36], [37]. The fifth perfect number was rediscovered by Regiomontanus during his stay at the University of Vienna, which he left in 1461 (see [235]). It has also been found in amanuscript written by an anonymous author around 1458, while the fifth and sixth perfect numbers have been found in another manuscript by the some author, shortly after 1460. The fifth perfect number is 33550336, and the sixth is 8589869056. These show that Nicomachus’ first claims (a) and (c) are false, since the fifth perfect number has 8 digits, and the fifth and sixth perfect numbers both ended in 6. In 1536 Hudalrichus Regius published ”Utriusque Arithmetices” in which he found the first prime p (p = 11) such that 2p−1(2p − 1) = 2047 = 23 · 89 is not perfect. In 1603 Cataldi found the factors of all numbers up to 800 and also a table of all primes up to 750. He used his list of primes to check that 219 − 1 = 524287 was prime, so he had found the seventh perfect number 137438691328. Among the many mathematicians interested in perfect numbers one should mention Descartes, who in 1638 in a letter to Mersenne wrote ([72]): ”... Perfect numbers are very few... as few are the perfect men...” In this sense see also a Persian manuscript [331]. ”I think I am able to prove there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort... But, whatever method one might use, it would require a great deal of time to look for these numbers...” The next major contribution was made by Fermat [225], [313]. He told Roberval in 1636 that he was working on the topic and, although the problems were very difficult, he intended to publish a treatise on the topic. The treatise would never been written, perhaps because Fermat didn’t achieve the substantial results he had hoped. In June 1640 Fermat wrote a letter to Mersenne telling him about his discoveries on perfect numbers. Shortly after writing to Mersenne, Fermat wrote to Frenicle de Bessy, by generalizing the results in the earlier letter. In his investigations Fermat used three theorems: (a) if n is composite, then 2n − 1iscomposite; (b) if n is prime, an − a is multiple of n;(c)ifn is prime, p is a divisor of 2n − 1, then p − 1isa multiple of n.

18 PERFECT NUMBERS

Using his ”Little theorem”, Fermat showed that 223 − 1was composite and also 237 − 1iscomposite, too. Mersenne was very interested in the results that Fermat sent him on perfect numbers. In 1644 he published ”Cogitata physica mathematica”, in which he claimed that 2p−1(2p − 1) is perfect for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257 and for no other value of p up to 257. It is remarkable that among the 47 primes p between 19 and 258 for which 2p − 1isprime, for 42 cases Mersenne was right. Primes of the form 2p − 1 are called Mersenne primes. (For a recent table of known Mersenne primes, see the site http://www.utm.edu/research/primes/mersenne/index.html). The next mathematician who made important contributions was Euler. In 1732 he proved that the eighth perfect number was 230(231 − 1).Itwas the first seen perfect number discovered for 125 years. But the major contributions by Euler were obtained in two unpublished manuscripts during his life. In one of them he proved the converse of Euclid’s statement: All even perfect number are of the form 2p−1(2p − 1). (2) By quoting R. C. Vaughan [314]: ”we have an example of a theorem that took 2000 years to prove... But pure mathematicians are used to working on a vast time scale...” Euler’s results on odd perfect numbers and amicable numbers will be considered later. After Euler’s discovery of primality of 231 − 1, the search for perfect numbers had now become an attempt to check whether Mersenne was right with his claims. The first error in Mersenne’s list was discovered in 1876 by Lucas, who showed that 267 − 1iscomposite. But 2127 − 1isaMersenne prime, so he obtained a new perfect number (but not the nineth, but as later will be obvious the twelfth). Lucas made also a theoretical discovery too, which later modifed by Lehmer will be the basis of a computer search for Mersenne primes. In 1883 Pervusin showed that 261 − 1isprime, thus giving the nineth perfect number. In 1911, resp. 1914 Powers proved that 289 − 1 resp. 2101 − 1 are primes. 288(289 − 1) wasinfact the last perfect number discovered by hand calculations, all others being found using theoretical elements or a computer. In fact computers have led to a revival of interest in the discovery of Mersenne primes, and therefore of perfect numbers. The first significant results on odd perfect numbers were obtained by Sylvester. In his opinion (see e.g. [317]): ”... the existence of an odd perfect number its escape, so to say, from the complex web of conditions which hem it in on all sides - would be little short of a miracle...”

19 CHAPTER 1

In 1888 he proved that any odd perfect number must have at least 4 distinct prime factors. Later, in the same year he improved his result for ﬁve factors. The developments, as well as recent results will be studied separately. Finally, we shall include here for the sake of completeness all known perfect numbers along with the year of discovery and the discoverer (s). Let Pk be the kth p−1 p p perfect number. Then Pk = 2 (2 − 1) = A p, where 2 − 1isaMersenne prime. P1 = A2 = 6, P2 = A3 = 28, P3 = A5 = 496, P4 = A7 = 8128, P5 = A13 (1456, anonymous), P6 = A17 (1588 Cataldi), P7 = A19 (1588, Cataldi), P8 = A31 (1772, Euler), P9 = A61 (1883, Pervushin), P10 = A89 (1911, Powers), P11 = A107 (1914, Powers), P12 = A127 (1876, Lucas), P13 = A521 (1952, Robinson), P14 = A607 (1952, Robinson), P15 = A1279 (1952, Robinson), P16 = A2203 (1952, Robinson), P17 = A2281 (1952, Robinson), P18 = A3217 (1957, Riesel), P19 = A4253 (1961, Hurwitz), P20 = A4423 (1961, Hurwitz), P21 = A9689 (1963, Gillies), P22 = A9941 (1963, Gillies), P23 = A11213 (1963, Gillies), P24 = A19937 (1971, Tuckerman), P25 = A21701 (1978, Noll and Nickel), P26 = A23209 (1979, Noll), P27 = A44497 (1979, Nelson and Slowinski), P28 = A86243 (1982, Slowinski), P29 = A110503 (1988, Colquitt and Welsh), P30 = A132049 (1983, Slowinski), P31 = A216091 (1985, Slowin- ski), P32 = A756839 (1992, Slowinski and Gage), P33 = A859433 (1994, Slowinski and Gage), P34 = A1257787 (1996, Slowinski and Gage), P35 = A1398269 (1996, Ar- mengaud, Woltman, et al. (GIMPS)), P36 = A2976221 (1997, Spence, Woltman, et al. (GIMPS)), P37 = A3021377 (1998, Clarkson, Woltman, Kurowski et al. (GIMPS, Primenet)), P38 = A6972593 (1999, Hajratwala, Woltman, Kurowki et al. (GIMPS, Primenet)), P?? = A13466917 (2001, Cameron, Woltman, Kurowski, et al.). Recently, M. Shafer (see http://mathworld.wolfram.com) has announced the discovery of the 40th Mersenne prime, giving A20996011. The full values of the ﬁrst seventeen perfect numbers are written also in the note by H. S. Uhler [312] from 1954. Foraquick perfect number analyzer by Brendan McCarthy see the applet at http://ccirs.camosun.bc.ca/∼jbritton/jbperfect/htm. The 39th Mersenne prime is of course, a very large prime, having a number of 4053946 digits (it seems that it is the largest known prime). There have been discovered also very large primes of other forms. For example, in 2002, Muischnek and Gallot discovered the prime number 105747665536 + 1. For the largest known primes of various forms see the site: http://www.utm.edu/research/primes/largest.html.

1.3 Even perfect numbers

Let σ(n) denote the sum of all positive divisors of n. Then n is perfect if σ(n) = 2n (1)

20 PERFECT NUMBERS

As we have seen in the Introduction, all known perfect numbers are even, and by the Euclid-Euler theorem n can be written as

n = 2k−1(2k − 1), (2) where 2k − 1isaprime (called also as ”Mersenne prime”). Actually k must be prime. The ﬁrst two perfect numbers, namely 6 and 28 are perhaps the most ”human” since are closely related to our life (number of days of a week, of a month, of a woman cycle, etc.). The famous mathematician and computer scientist D. Knuth in his interesting homepage (http://www-cs- faculty.stanford.edu/∼knuth/retd.htm) on the occasion of his retirement says: ”... I’m proud of the 28 students for whom I was a dissertation advisor (see vita); and I know that 28 is a perfect number...” 28 is in fact the single even perfect number of the form

x3 + 1 (3)

(x positive integer), proved by A. Makowski [205]. As corollaries of this fact Makowski deduces that the single even perfect number of the form nn + 1is28; (4) and that there is no even perfect number of the form

...n nn + 1,(5) where the number of n’s is ≥ 3. By generalization, A. Rotkiewicz [263] proves that 28 is the single even perfect number of the form

an + bn, where (a, b) = 1andn > 1.(6)

If n > 2and(a, b) = 1, he proves also that there is no even perfect number of the form an − bn.(7) Not being aware of Rotkiewicz’s result (6), T. N. Sinha [288] has rediscovered it in 1974. The numbers 6 and 28 are the only couple of perfect numbers of the form n − 1 n(n + 1) and , where n > 1; and this is obtained for n = 7. This is a result of L. 2 Jones [157]. (8) We note that this result applies the Euclid-Euler theorem (representation of even perfect numbers), as well as a theorem by Euler on the form of odd perfect numbers (see the next section).

21 CHAPTER 1

As for the digits of even perfect numbers, as already Nicomachus (see section 2) remarked, the last digits are always 6 or 8 (but not in alternate order, as he thought), proved rigorously by E. Lucas in 1891 (see [84], p. 27, and also [138]). (9) Let us now sum the digits of any even perfect number (except 6), then sum the digits of the resulting number,..., etc., repeating this process until we get a single digit. Then this single digit will be one. (10) See [332] for this result, with a proof. Let A(n) be the set of prime divisors of n > 1. If n is an even perfect number, then it is immediate that A(n) = A(σ (n)). (11) Now, an interesting fact, due to C. Pomerance [240] states that, reciprocally, if (11) holds true for a number n, then n must be even perfect. For the additive representation of even perfect numbers, by an interesting result by R. L. Francis [103] any even perfect number > 28 can be represented as the sum of at least two perfect numbers. (12) For the values of other arithmetic functions at even perfect numbers we quote the following result of S. M. Ruiz [264]: Let S(n) be the Smarandache function, deﬁned by

S(n) = min{k ∈ N : n|k!}.(13)

Then if n is even perfect, then one has

S(n) = Mp (14)

p where Mp = 2 − 1isthe Mersenne prime in the known form of n.For a simple proof, see J. Sandor´ [272]. In [272] there is proved also that S(Mp) ≡ 1 (mod p), and that if 22p + 1isprime too, then S(24p − 1) = 22p + 1 ≡ 1 (mod 4p). For the values of Euler’s function on even perfect numbers, see S. Asadulla [8]. For perfect numbers concerning a Fibonacci sequence, see [234]. F. Luca [197] proved also that there are no perfect Fibonacci or Lucas numbers. In [198] he proved that there are only ﬁnitely many multiply perfect numbers in these sequences. K. Ford [102] considered numbers n such that d(n) and σ(n) are both perfect numbers (called ”sublime numbers”). There are only two known such numbers, namely n = 12 and n = 2126(261 − 1)(231 − 1)(219 − 1)(27 − 1)(25 − 1)(23 − 1).Itisnot known if any odd sublime number exists. D. Iannucci (see his electronic paper ”The Kaprekar numbers”, J. Integer Se- quences, 3(2000), article 00.1.2) proved that every even perfect number is a Kaprekar 2 number in the binary base, e.g. (28)2 = 11100 and 1100 = 1100010000 with 100 + 010000 = 11100 (703 in base 10 is Kaprekar means that 7032 = 494209 where 494 + 209 = 703).

22 PERFECT NUMBERS

We wish to mention also some new proofs of the Euler theorem on the form of an even perfect number. In standard textbooks, usually it is given Euler’s proof, in a slightly simpliﬁed form given by L. E. Dickson in 1913 ([82], [80]). An earlier proof was given by R. D. Carmichael [43]. A new proof has been published by Gy. Kisgergely [174] in a paper written in Hungarian. Another proof, due to J. Sandor´ ([271], [273]) is based on the simple inequality

σ(ab) ≥ aσ(b)(15) with equality only for a = 1. This method enabled him also to obtain a new proof on the form of even superperfect numbers (see later). Foreven perfect numbers see a paper by S. J. Bezuska and M. J. Kenney [24] (where 36 perfect numbers are mentioned (up to 1997!)). See also G. L. Cohen [56].

1.4 Odd perfect numbers

There is a good account of results until 1957 in the paper by P. J. McCarthy [46]. The ﬁrst important result on odd perfect numbers was obtained by Euler [98] when he proved that such a number n should have the representation

α β β = 2 1 ... 2 r ( ) n p q1 qr 1 where p, qi (i = 1, r) are distinct odd primes and p ≡ 1 (mod 4), α ≡ 1 (mod 4). Here pα is called the Euler factor of n. Another noteworthy result, mentioned also in the Introduction is due to J. J. Sylvester [310] who proved that we must have r ≥ 4 and that r ≥ 5if n ≡ 0 (mod 3). (2) The modern revival of interest in the problem of odd perfect numbers seems to have been begun by R. Steurwald [296] who proved that n cannot be perfect if

β1 = ···=βr = 1.(3) A. Brauer [33] and H.-J. Kanold [159] proved the same if

β1 = 2andβ2 = ···=βr = 1 (4) In [158] Kanold proved that n is not perfect if

β1 = ···=βr = 2 (5) and also if the

2βi + 1 (i = 1, r) have as a common factor 9, 15, 21, or 33. (6)

23 CHAPTER 1

Similarly, he proved [165] that n cannot be perfect if β1 = β2 = 2, β3 = ··· = βr = 1; and if α = 5 and βi = 1or2(i = 1, r).(7) He obtained similar results in [161]. P. J. McCarthy [47] proved that if n is perfect and prime to 3, and β2 = ···=βr = 1, then

q1 ≡ 1 (mod 3)(8)

In 1971 W. L. McDaniel [76] improved result (6) by proving that 3 cannot be a common divisor of 2βi + 1 (i = 1, r).(9) A year later, P. Hagis, Jr. and W. L. McDaniel [133] showed that n cannot be perfect if β1 = ···=βr = 3.(10) We note here that in the above papers the theory of cyclotomic polynomials, as well as diophantine equations are widely used. Some computations use modern computers, too. It is of interest to note that in the proof of (10) the following results due to U. Kuhnel¨ [184], resp. H.-J. Kanold [158] are used: If n is odd perfect, then 105 n;(11) If n is odd perfect, and if in (1) s is a common factor of 2βi + 1 (i = 1, r), then s4|n.(12) Another reﬁnement of Euler’s theorem is due to J. A. Ewell [99]. If the prime q1,...,qr (together with their corresponding exponents) are relabeled as p1, p2,...,pk, h1, h2,...,ht so that pi (i = 1, k) are of the form ≡ 1 (mod 4) and each h j ( j = 1, t) are of the form ≡−1 (mod 4),say

k t α β = α 2 i 2 j ( ) n p pi h j 13 i=1 j=1 then: (i) the ordered set A(k) ={α1,...,αk} contains an even number of odd numbers, provided α and p belong to the same residue class (mod 8); (ii) A(k) contains an odd number of odd numbers, provided that α and p belong to different classes (mod 8).(14) Of a similar nature are also the following theorems due to G. L. Cohen and R. J. Williams [67]: If n is an odd perfect number, then in (1) if we assume that β1 = ··· = βr = β, then we must have β ∈{6, 8, 11, 14, 18}; (15)

If n is as above and β1 = ···=βr = 1, then

β1 ∈{5, 6}; (16)

24 PERFECT NUMBERS

They obtain also (independently) a simpler proof of result (14). The theorem of Sylvester on the lower bounds for r in (1) has been improved by several authors. I. S. Gradstein [110], U. Kuhnel¨ [184] and G. C. Webber [324] proved that in (2) r ≥ 5 holds without any condition. (17) Recall, that r = ω(n) − 1. The result r ≥ 6 (18) has been discovered by N. Robbins [259] and C. Pomerance [237]; while the strongest result known today, namely r ≥ 7 (19) is due to P. Hagis, Jr. [120] and J. E. Z. Chein [51]. Foranew, algorithmic approach, see G. L. Cohen and R. M. Sorli [On the number of distinct prime factors of an odd perfect number, J. Discr. Algorithms 1(2003), 21- 35]. Result (2) of Sylvester has been sharpened to

r ≥ 8ifn ≡ 0 (mod 3)(20) by H.-J. Kanold [160] and to

r ≥ 10 if n ≡ 0 (mod 3)(21) by M. Kishore [179]. P. Hagis, Jr. [123] proved the same for (n, 3) = 1. For a survey of earlier results, see D. S. Mitrinovic-J.S´ andor´ [218] (see p. 100-102). E. Catalan proved in 1888 (see [84]) that if an odd perfect number is not divisible by 3, 5 or 7, it has at least 26 distinct prime factors, and thus has at least 45 digits. T. Pepin showed in 1897 (see [84]) that an odd perfect number relatively prime to 3 · 7, 3 · 5or3· 5 · 7 contains at least 11, 14, or 19 distinct prime factors, respectively; and cannot have the form 5(mod6). In the above mentioned paper [160] Kanold proved also that an odd perfect number must have at least a prime divisor greater than or equal to 61. (22) J. B. Muskat [223] in 1966 proved that an odd perfect number n is divisible by a prime power > 1012.(23) C. Pomerance [238] proved that the second largest prime divisor of n must be at least 139. (24) and in 1980 P. Hagis, Jr. [121], improved this to 1000. (25) Related to the largest prime factor of an odd perfect number n, this is greater than

100129 (26)

25 CHAPTER 1 as shown by P. Hagis, Jr. and W. J. Daniel [134]; and improved to

300000 (27) by J. T. Condict [68]. Result (25) on the second largest prime factor has been recently improved to

10000 (28) by D. E. Iannucci [150]. Ianucci further [151] obtained for the third largest prime factor the lower bound 100 (29) Iannucci and Sorli [153] proved that if n is odd perfect, then (n) ≥ 37, (n) being the total number of prime factors of n. Kanold’s result (22) was subsequently improved. As a corollary of a general result (which we shall state later) N. Robbins [260] proved that if an odd perfect number n is divisible by 17, then n must have a prime factor not smaller than 577. (30) The strongest result was due to P. Hagis, Jr. and G. L. Cohen [132] who obtained, without any condition the lower bound

106 (31)

Very recently, this has been improved to 107 by P. M. Jenkins [155]. See also A. Grytczuk and M. Wojtowicz´ [113] on the magnitude of perfect numbers. Related to the lower bounds of odd perfect numbers, a substantial result was obtained in 1973 by P. Hagis, Jr. [119], giving the lower bound 1050. This was subsequently improved to 10160 by R. P. Brent and G. L. Cohen [34], to 10300 by Brent, Cohen and H. J. J. te Riele [35]. In a recent paper by S. Davis [79] a rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such that σ(n) could be equal to 2n,where n belongs to a ﬁxed interval with a lower limit greater than 10300. Bounds of a general nature were ﬁrst considered by Kanold. In [158] he proved that if n given by (1) is odd perfect, then the largest prime divisor of n is greater then

2 · max{α + 1, 2βi + 1: i = 1, r} (32) In [160] he proved that if n is odd perfect and p is a Gaussian prime, then

26 PERFECT NUMBERS

Suppose n is odd perfect and that(p − 1, 2βi + 1) = 1, i = 1, r.Leta be as 3a(a − 1) β above. Then 1 < a < r and α ≤ min a(a − 2), , when pa−1 σ(q2 i ) 4 i (a − 1)(3a − 2) for any i = 1, r; and α ≤ , otherwise. (34) 4 Acomplicated lower bound has been given by N. Robbins [260]. Suppose u is an odd prime. Let a1(u)

L(u) = min{a1(u), b1(u), c1(u)}.(35)

Now, if n given by (1) is an odd perfect number and u is a Fermat prime such that um||n, then M(n) ≥ L(u)(36) where

M(n) = max{p, q1,...,qr } (37)

Foranexample, the values of an(3), an(5), an(17), bn(3), bn(5), bn(17), cn(3), cn(5), cn(17), consider the tables:

n 1 2 3 4 5 6 an(3) 17 89 197 233 449 521 an(5) 149 349 449 1249 1549 1949 an(17) 577 1733 8669 14449 19073 22541

27 CHAPTER 1

n 1 2 3 4 5 6 bn(3) 19 61 67 79 97 127 bn(5) 1381 2221 3221 31 bn(17) < 10

n 1 2 3 4 5 6 cn(3) 271 821 919 1021 1087 1093 cn(5) 400291 31 cn(17) < 10

Various bounds for the prime factors of an odd perfect number have been deduced. Let p1 be the least prime factor of n given by (1). Then Cl. Servais [283] proved ﬁrst that

p1 ≤ r + 1 = ω(n) = A (38) where ω(n) = A denotes the number of distinct prime factors of n.For a recent extension of the Servais result see [23]. In 1952 O. Grun¨ [112] improved this to

2 p < A + 2 (39) 1 3 The bound (39) was deduced also independently by H. Salie[´ 268] in 1953, as well as M. Perisastri [233] in 1958. We note that (39), or even (38) are many times rediscovered by different authors (see e.g. [86] for a weaker result than (38)). By using some general results concerning systems of Diophantine equations, H.-J. Kanold [162] in 1952 proved that if n is odd q + 1 perfect (or even multiply perfect), then n has a square factor ≥ ,(40) 2 where q is the largest divisor of n (denoted also as q = P(n)). A = ai Let an odd perfect number n be written in the prime factorization n pi . i=1 Then 2i(i+1)/2 pi <(4A) , i = 1, A (41) which is due to C. Pomerance [239]. For 2 ≤ i ≤ 6 one has the improvement

2i−1 pi < 2 (r − i + 1)(42) due to M. Kishore [177].

28 PERFECT NUMBERS

D. R. Heath-Brown [143] proved that if an odd perfect number n has A distinct prime factors, then A n < 44 (43) 1 Various bounds for the sum S = (where p is prime) for an odd perfect p/n p number have been obtained by D. Suryanarayana [304] and P. Hagis Jr. and D. Surya- narayana [137]. For example, when n ≡ 1 (mod 12) and 5|n, then 1 0.64738 < S < + log 50/31 = 0.67804 (44) 5 If n ≡ 1 (mod 12),but5 n, then

0.66745 < S < 0.69315 (45)

See Mitrinovic-S´ andor´ [218] (pp. 103-104) for more detailed results. In D. Surya- p P = n narayana [305] similar results are obtained for − .Namely, e.g., if is p/n p 1 odd perfect and n ≡ 1 (mod 12),5|n, then

2 < P < 2.031002; (46)

If n ≡ 1 (mod 12),5 n, then

2 < P < 2.014754,(47) etc. The above congruence n ≡ 1 (mod 12) is not considered at random. A theorem by J. Touchard [311] states that for an odd perfect number n one has

n ≡ 1 (mod 12), or n ≡ 9 (mod 36)(48)

The initial proof given by Touchard was quite complicated, for an elementary proof see a recent paper by J. A. Holdener [144]. An earlier proof was obtained by M. Satyanarayana [280]. A somewhat later one is by M. Raghavachari [246]. An immediate application of Touchard’s theorem (48) and the fact that 2p−1(2p − 1) ≡ 4 (mod 12) for any prime p ≥ 3, is that two consecutive positive integers cannot be both perfect numbers (see F. Luca [195]). (49) Another famous result is due to L. E. Dickson [82]: There are only a ﬁnite number of odd perfect numbers with a given number of prime divisor. This may be reformu- ={ = a1 ... ar ,..., lated more explicitly as follows: The set D n p1 pr : p1 pr given odd primes; ai = 0, 1, 2,...} contains at most a ﬁnite set of perfect numbers. (50)

29 CHAPTER 1

New proofs of this theorem were obtained e.g. by H. N. Shapiro [284], H.-J. Kanold [167], R. W. Van der Waall [316]. In fact Shapiro, in the paper [284], and in a new paper [285] obtains a very large generalization of Dickson’s theorem. Kanold’s method works also for k-multiperfect numbers (see [168]), another proof appears also in [316]. In 1990 Kanold [173] obtains a new generalization of Dickson’s theorem. We now return to the representation (1) of an odd perfect number n,inorderto consider certain result of a new type concerning the term pα (”Euler factor”) or the βi other terms qi . β σ( α)/ σ( 2 i ) In 1947 R. J. Levit [189] proved that n is not perfect if p 2 and qi (i = 1, r) are prime powers. (51) Recently P. Starni [294] proved that if n is an odd perfect number with qi ≡ 3 (mod 4)(i = 1, r), then σ(pα)/2isacomposite number. (52) Let n be written as n = pα M2 where (see (1)) as usual p ≡ 1 ≡ 1 (mod 4), (p, M) = 1. In 1993 P. Starni [295] proved that if α + 2isprime or pseudoprime in base p and (α + 2, p − 1) = 1, then

M2 ≡ 0 (mod α + 2). (53) p − 1 As a corollary, if n = pα M2 is odd perfect, α + 2 = prime, and α + 2 > , 4 then M ≡ 0 (mod α + 2)(54) r β α < σ( 2 i ) J. Slowak [291] shows that p qi ,soinfact one has the following i=1 representation of an odd perfect number: σ(pα) n = pα d, d > 1 (55) 2 Some words on the density of odd perfect numbers: In 1954 Kanold [166] proved that this density is 0. This follows also from a result by B. Volkmann [315]. Stronger results have been obtained in the following years by B. Hornfeck [146] who showed that for V (x) = card{n ≤ x : n perfect} (56) one has V (x)

30 PERFECT NUMBERS

Kanold [169] improved these to

x1/4 log x V (x)convergent series. Finally, we wish to point out some new issues on odd perfect number, along with some open problems. By writing n = pα M2,where

p ≡ α ≡ 1 (mod 4), (62) clearly if one has σ(M2) = pα,σ(pα) = 2M2,(63) then n is odd perfect. In 1977 D. Suryanarayana [309] raised the question, that should every odd perfect number (which necessarily has the form (62)) also necessarily satisfy the relations (63)? This was answered in the negative by G. G. Dandapat, J. L. Hunsucker and C. Pomerance [74], and also later by E. Z. Chen utilizing a deep result of Ljunggern. D. Suryanarayana [309] raised also the following problem. If n = pα M2 is an odd perfect number so that p ≡ α ≡ 1 (mod 4) and (p, M) = 1, does it necessarily follow that there exist a divisor d|M such that

σ(d2) = pα M2/d2 and σ(pα M2/d2) = 2d2? (64)

This problem is still open. Another open problem by Suryanarayana is that: is it true that every odd perfect number is of the form mσ(m) for some odd integer m;ifso, is (m,σ(m)) = 1 necessarily? (65)

31 CHAPTER 1

M. V. Subbarao [301] raises the problem: Does every odd perfect number n (if such exist) have the representation 1 n = mσ(m)? (66) 2 Another question: whenever n given by (66) is perfect, does it follow that n is odd and (m,σ(m)) = 1? (67) C. W. Anderson (see [239]) has conjectured that the set of rational numbers that arise as σ(n)/n for some n,isarecursive set. (68)

1.5 Perfect, multiperfect and multiply perfect numbers

First we wish to mention that the notion of a perfect number has been extended to Gaussian integers (see R. Spira [292], W. L. McDaniel [77], M. Hausman [141], D. S. Mitrinovic–J.S´ andor´ [218]) to real quadratic fields (see E. Bedocchi [15]), or generally to unique factorization domains (see W. L. McDaniel [78]). A number n is called multiperfect,ifthere exist a positive integer k ≥ 1such that σ(n) = kn (1) In this case n is called also as k-perfect number. The union of all k-perfect numbers when k ∈ N is the set of multiply perfect numbers. Thus n is called multiply perfect when n|σ(n)(2) In the sections with perfect numbers (when k = 2in(1)) we sometimes mentioned that some results which were true for perfect numbers do hold also for multiperfect numbers. (For the history of multiply perfect and multiperfect numbers up to 1907, see Dickson [84].) For example, the famous Dickson theorem 4.(50) holds true also for k-perfect number, as proved by H.-J. Kanold [168]. A positive integer n is called primitive if cannot be written in the form m = st, where s is an even perfect number and (s, t) = 1. Kanold proved that for any k there are only a finitely many primitive k-perfect numbers with a fixed number of distinct prime factors. (3) C. Pomerance [239] proved that for every k ≥ 1(even rational number) and every non-negative integer K , there is an effectively computable number N(k, K ) such that if ω(n) = K and n is primitive k-perfect, then

n ≤ N(k, K ). (4)

32 PERFECT NUMBERS

Here ω(n) denotes, as usual, the number of distinct prime divisors of n. Let A(n) be the set of all prime divisors of n.In1998 W. Carlip, E. Jacobson and L. Somer [42] proved the following theorem: For ﬁxed integers k and t, there exist at most ﬁnitely many squarefree integers n such that

card A(N) ≤ t and n is k − perfect. (5)

P. J. McCarthy [48] has shown that if n is k-perfect, then

ω(n) ≥ k2 − 1 (6)

For some improvements, see W. L. McDaniel [75]. McDaniel shows also that there exists no odd k-perfect numbers with k odd of the form mα,where m is an integer, and α + 1isaprime, the square of a prime, or the cube of a prime. (7) It is not known if there exist odd k-perfect numbers, but actually up to recently (February 2002) we have a total of 5040 known k-perfect numbers. There are 39 2-perfect (i.e. classical perfect), 6 3-perfect, 36 4-perfect, 65 5-perfect, 245 6-perfect, 516 7-perfect, 1134 8-perfect, 923 10-perfect and 1 11-perfect numbers. For details regarding the discoverers see the site http://www.homes/uni- bielefeld.de/achim/mpn.html by A. Flammenkamp. P. Hagis Jr. and G. L. Cohen [130] have shown that the largest prime factor of a k-perfect number is at least 100129, (8) while the second largest prime factor is

≥ 1009.(9)

Later, Hagis Jr. [125] proved that the third largest divisor is

≥ 101 (10)

The density of k-perfect numbers is 0, as ﬁrst was shown by Kanold, but in 1957 Hornfeck and Wirsing proved that

P(x) = o(xε)(11) for any ε>0, where P(x) denotes the number of k-perfect numbers not exceeding x.Infact in the book by Erdos¨ and Graham [92] one can read that Wirsing can show that P(x) 0 constants. (Here k > 0 can be even rational number.) 1 Various bounds for S = in the case of an odd perfect number have been p/n p included in section 4 (see 4.44). G. L. Cohen [55] in 1980 proved the following:

33 CHAPTER 1

Let n be k-perfect. Then   , log k log k if k is odd < S < log k (13) 3 log 3/2  , if k is even 3 log 4/3 for n odd; and 1 log k/2 1 log 2k/3 + < S < + (14) 2 3 log 3/2 2 3 log 4/3 log k if n is even. These improve S > , due to A. Krawczyk [183]. 2 M. Bencze [19] proved that

r( r 3/2 − 1)

~~ω(n) ≥ 9 (17)~~

~~E. A. Bugulov [39] showed in fact~~

~~ω(n) ≥ 11; (18) this has been rediscovered by M. Kishore [180]. In 1983 H. Reidling [248] showed~~

~~ω(n) ≥ 12,(19) and this has been again rediscovered by Kishore [181]. In 1982 W. E. Beck and R. M. Najar [14] obtained for an odd triperfect number n the lower bound n > 1050 (20)~~

~~34 PERFECT NUMBERS~~

This was improved to n > 1060 (21) by L. B. Alexander [4], and to n > 1070 (22) by Cohen and Hagis, Jr. [130]. As we mentioned, an odd triperfect number must be a square: D. Iannucci [152] proves that the square root of an odd triperfect number cannot be squarefree number. (23) The important theorem by Touchard (see 4.48) can be extended in some cases also to multiperfect numbers. J. A. Holdener [145] proves that if n is odd k-perfect, where k is not divisible by 3 or 4, then n ≡ 1 (mod 12) or n ≡ 9 (mod 36). (24) Now certain facts on multiply perfect numbers. (Remark that many authors con- fuse the words ”multiperfect” and ”multiply perfect”. Clearly if n is k-perfect, then it is multiply perfect. But the converse is not true.) R. D. Carmichael [44] found all multiply perfect numbers less than 109. These are: 1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240 (this corrected list appears in [9]). (25) Since, as we mentioned before, there are more than 5040 multiperfect numbers, and these are also multiply perfect numbers, today (in the computer era) we know multiply perfect numbers having about 10000 digits. However, it is not known whether or not there are infinitely many such numbers. See also R. K. Guy [114]. The estimate 4.(60) for the number of perfect numbers ≤ x holds true essentially also for multiply perfect numbers, so for these numbers m(x) one has c log x log log x log x m(x) = O exp (26) log log x due to Hornfeck and Wirsing [148]. We now study some miscellaneous results related to multiply perfect numbers. H. Harboth [140] takes in place of σ(n) the value S(n) = the sum of all possible sums of distinct divisors of n (in fact one obtains S(n) = σ(n) · 2d(n)−1, where d(n) is the number of divisors of n). She proves that n|S(n) for infinitely many n (27) thus the open question above is solved in the case considered here. In a difficult proposed problem, C. Pomerance [242] considers multiply perfect numbers of the form n!, and shows that this is possible only for n ∈{1, 3, 5}.(28)

~~35 CHAPTER 1~~

L. Cheng [52] has proved that n = 23 · 3 · 5, 25 · 3 · 7 and 25 · 33 · 5 · 7 are the only integers satisfying σ(n) = ω(n) · n.(29) For multiply perfect numbers in Lucas sequences see [194]. Finally, let us stop at certain problems relating complexity theory of algorithms. Gill’s complexity class ([107]) BPP is the class of languages recognized in polynomial time by a probabilistic Turing machine, with two-sided error probability bounded by a constant away from 1/2. Now Theorem 8 of [9] states that the set of multiply perfect numbers is in BPP. (30) Bach, Miller and Shallit prove the similar result for perfect numbers, as well as amicable numbers. (31) For algorithmic number theory, see [10].

~~1.6 Quasiperfect, almost perfect, and pseudoperfect numbers~~

A number n is called quasiperfect,if σ(n) = 2n + 1 (1) No such numbers are known, but necessary conditions have been obtained by various authors. P. Cattaneo [50] proved that if n is quasiperfect and r|σ(n), then r ≡ 1or3 (mod 8)(2) If p2α||n and q|(2α + 1), then (q − 1)(p + 1) ≡ 0or4 (mod 16)(3) Cattaneo proved also that an odd quasiperfect number must be a perfect square. (4) H. L. Abbott, C. E. Aull, E. Brown and D. Suryanarayana [1] as well as G. L. Cohen [57] deduced other conditions. For example, in [57] it is shown that there is no quasiperfect number of the forms m4 or m6 (in this last case one suppose also (m, 3) = 1) (5) It is shown also that there are no quasiperfect numbers of the form 32am2b where 3 m, a ≡ 2 (mod 8) and b ≡ 0 (mod 5) or b ≡ 0 (mod 11)(6) r 6qi +2 If a number of the form pi (pi primes) is quasiperfect, then i=1 r ≥ 230876 (7)

~~36 PERFECT NUMBERS~~

~~Abbott, Aull, Brown and Suryanarayana [1] showed that a quasiperfect number n must have at least ﬁve distinct divisors and~~

~~n > 1020 (8)~~

~~M. Kishore [175] improved this to~~

~~ω(n) ≥ 6 and n > 1030 (9) while Cohen and Hagis, Jr. [61] obtained the reﬁnement~~

~~ω(n) ≥ 7 and n > 1035 (10)~~

~~When 3 n, this has been earlier obtained by Jerrard and Temperley [156]. Kanold [172] proved that there are only ﬁnitely many quasiperfect numbers n for which ω(n)~~

~~σ(n) = 2n + 2~~

If 2k − 3 = prime, then n = 2k−1(2k − 3) is a solution. We note that 2k − 3 are primes e.g. for k = 2, 3, 4, 5, 6, 9, 10, 12, 14, 20, etc. A solution of other type is n = 650. We now introduce almost perfect numbers. A number n is called so, if one has

~~σ(n) = 2n − 1 (17)~~

~~37 CHAPTER 1~~

Again, we should note that in the literature ”almost perfect” sometimes means quasiperfect in the sense of (1), or almost perfect, in the sense of (17) (see e.g. [156], [71]). S. Singh [287] calls almost perfect numbers ”slightly defective” while quasiperfect numbers ”slightly excessive”. It is easy to check that powers of 2 satisfy (17); no other almost perfect numbers are known. An inﬁnite class of numbers which are not almost perfect is given by J. T. Cross [71] as follows: Let p denote an odd prime. If 2m+1 > p, then no multiple of 2m p is almost perfect. (18) In 1978 M. Kishore [176] proved that if n is an odd almost perfect number, then

~~ω(n) ≥ 6 (19)~~

Cross [71] notes that the argument by Jerrard and Temperley [156] can be modi- ﬁed to show that if 3 n, then ω(n) ≥ 7 (20) for such numbers. An analogous result to (16) in case of almost perfect numbers is

2i−1 pi < 2 (r − i + 1), 2 ≤ i ≤ 5 (21) and p6 < 23775427335(r − 5)(22) due to Kishore [178]. An odd almost perfect number n should be a perfect square, a1 || = ≥ and if p1 n, where p1 3istheleast prime divisor of n, then a1 12. (23) For this result, see also [176].

~~1.7 Superperfect and related numbers~~

~~A number n is called superperfect (after D. Suryanarayana [307]) if~~

~~σ(σ(n)) = 2n (1)~~

~~Suryanarayana and Kanold [171] have determined the general form of even superperfect numbers. These can be written as~~

~~n = 2k (k ≥ 1), where 2k+1 − 1isprime (2)~~

~~New proofs of (2) have been obtained by G. Lord [193] and J. Sandor´ ([271], [273]). The odd superperfect numbers are not known, but they must be perfect squares (3)~~

38 PERFECT NUMBERS as proved by Kanold [307]. According to Guy [114] (p. 65), Dandepat et al. proved that if n is odd superperfect, then n or σ(n) is divisible by at least three distinct primes. (4) Suryanarayana [308] shows that odd numbers of this type cannot have the form p2α (p prime). (5) Hunsucker and Pomerance [149] give the lower bound

~~7 · 1024 (6) for the odd superperfect numbers. D. Bode [26] obtained that the number of even superperfect numbers less than x is~~

~~o(log x/ log log x)(7)~~

~~By iterating the composition of σ,Bode [26] showed also that there are no even m-superperfect numbers for m ≥ 3, in the sense that~~

~~σ (m)(n) = 2n (8)~~

σ (m)( ) = σ(...σ( )...) where n n . mtimes Foranew proof, see Lord [193]. J. McCranie [70] has shown that there are no m-superperfect numbers less than 4.29 × 109 for any m ≥ 3. Sandor´ [270] considered the perfect number analogue for the composition of some arithmetic functions. He reproved (2) by showing that

~~σ(σ(n)) ≥ 2n for all even n (9) with equality only for n = 2k, with 2k+1 − 1 prime. Let ψ be the Dedekind arithmetic function, deﬁned by 1 ψ(n) = n 1 + (n > 1), ψ(1) = 1. p/n p~~

~~Then, an improvement of (9) is the inequality~~

σ(σ(n)) ≥ ψ(σ(n)) ≥ 2n for even n,(10) where the last equality holds only for n = 2k, with 2k+1 − 1prime. Superperfect number are called σ ◦ σ-perfect numbers, while numbers with property

~~ψ(σ(n)) = 2n (11)~~

39 CHAPTER 1 are called ψ ◦ σ-perfect numbers. All even perfect numbers of this type are determined. Equation (11) has at least an odd solution n = 3. Let n be an odd solution of (11). Then (see [270]) (a) n ≡−1 (mod 3), (b) n ≡ 7 (mod 12) (c) n ≡−4 (mod 21), n ≡ 10 (mod 21) (d) 2α · 3β σ(n) for all α, β ≥ 1. (12) Related to σ ◦ ψ-perfect numbers, i.e. satisfying

~~σ(ψ(n)) = 2n (13)~~

~~Sandor´ proves that the only even solution is n = 2. One has~~

m + 1 σ(ψ(n)) ≥ 2n for n even,(14) m with n = 2a, where m denotes the greatest odd divisor of n.Ifn is an odd σ ◦ ψ- perfect number, and p is an odd prime having the property σ(p +1)>2p, then p n (particularly 3, 5, 7, 11, 17, 19 n). (15) It is conjectured that (13) doesn’t have odd solutions. In connection to Bode’s result, Sandor´ [270] proves that for any m ≥ 3, there are no ψ ◦ ψ ◦ ···◦ψ − ( ) perfect numbers 16 m The problem of ”ψ-superperfect” numbers is completely settled as followed: The only solution of equation

~~ψ(ψ(n)) = 2n (17) is n = 3. But there are many ”ψ-triperfect” numbers as the even solutions of~~

ψ(ψ(n)) = 3n (18) are n = 2k,where k > 1. Let ϕ be the Euler totient. Sandor´ [270] proves that there are no ψ ◦ ϕ-perfect numbers which are odd. (19) If n is even, and the greatest odd divisor of n is ψ ◦ ϕ-deﬁcient, then n is also ψ ◦ ϕ-deﬁcient (i.e. (ψ ◦ ϕ)(n)<2n). Here it is conjectured also that

~~ψ(ϕ(m)) ≥ m for any odd m (20)~~

~~40 PERFECT NUMBERS~~

Related to ψ-multiperfect numbers the following result is true (Sandor´ [269]). We say that n is ω-multiple of m (see Ch. Wall [318]) if m|n and the sets of prime factors of m resp. n are identical. Let us now suppose that the least solution of equation ψ(n) = kn (21) is a squarefree number nk. Then all solutions of (21) are the ω-multiples of n.(22) Let us call n quasi-superperfect if

~~σ(σ(n)) = 2n + 1 (23)~~

Clearly the Mersenne primes are such; it is not known if there are others. Sub- barao [301] proves that if a prime p satisﬁes (23) then p is a Mersenne prime, otherwise σ(p) = 2r M2, where M has at leat two prime divisors. (24) Generally, if n satisﬁes (23), then n must be odd, and σ(a) must be of the form

~~σ(n) = 2r M2, where M is an odd integer. (25)~~

Subbarao proves also that there is no odd perfect number of the form n = 1 mσ(m) for which σ(m) is a power of n and with m quasi-superperfect (i.e. m 2 satisfying (23)). (26) He raises the following open problem: if m is quasi-superperfect, is the number 1 n given by n = mσ(m) a perfect number? (27) 2 The almost superperfect numbers are deﬁned by

~~σ(σ(n)) = 2n − 1 (28)~~

We cannot decide if there exist such numbers, in fact no reference to this question we were able to ﬁnd in the literature. (The analogous question, however, for unitary divisors, as we will see in a further section, is much easier). Another generalization of superperfect numbers has been considered by G. L. Cohen and H. J. J. te Riele [64]. They call a number n (m, k)-perfect if

~~σ (m)(n) = kn (29)~~

The classical perfect numbers are then the (1,2)-perfect, the k-multiperfect numbers are (1, k)-perfect, the superperfect numbers are (2,2)-perfect, the m-superperfect numbers are (m, 2)-perfect. They give a table of all (2, k)-perfect numbers up to 109. They gave a table for all (3, k)-perfect and (4, k)-perfect numbers up to 2 · 108 in

~~41 CHAPTER 1~~

1995 [63]. It is interesting to note that no new (1,3)-perfect numbers have been found in the last 350 years, nor any new (1,4)-perfect numbers in the last 70 years. For (2, k)-perfect numbers they prove the following general result: suppose l is an odd (2, k)-perfect number. For any a such that 2a|kσ(σ(2a)) and such that σ(2a) and σ(l) are relative prime, the number 2al is (2, 2−akσ(σ(2a)))-perfect. (30) Forexample, for p > 2, 2p−1 · 3 · 5 · 7 · 132 · 31 is (2,12)-perfect. They conjecture that all numbers n are (m, k)-perfect for some m, k; and verified it for all n ≤ 1000. (31) There are also some theoretical results supporting conjecture (31). Here one has to do with the properties relating to the iteration of σ.P.Erdos¨ [90] asked if (σ (m)(n))1/m has a limit as m →∞.Heconjectures that it is infinite for each n > 1. (32) Erdos,¨ Granville, Pomerance and Spiro [93] poses five new open problems, as follows: (i) For any n > 1, σ (m+1)(n)/σ (m)(n) → 1asm →∞ (33) (ii) For any n > 1, σ (m+1)(n)/σ (m)(n) →∞as m →∞ (34) (iii) For any n > 1, there is m with n|σ (m)(n) (35) (iv) For any n, l > 1 there is m with l|σ (m)(n) (36) (m1) (m2) (v) For any n1, n2 > 1, there are m1, m2 with σ (n1) = σ (n2).(37) Cohen and te Riele [64] give some computational evidence to indicate that state- ments (32), (34), (35), (36) are true, while (33) and (36) are false. A. Schinzel [281] asks if

lim inf σ (m)(n)/n < ∞ (38) n→∞ for each m,and observe that it follows for m = 2from a deep theorem of A. Renyi.´ A. Makowski and A. Schinzel [207] gave an elementary proof that for m = 2 the limit is 1. The result is also true for m = 3, proved by H. Maier [202]. (39) Makowski [206] showed earlier this result by assuming the inﬁnitude of Mersenne primes. He proves also that (38) is true if one assumes a general hypothesis H due to Schinzel [282]. (40) For the iteration properties of s(n) = σ(n)−n,aswell as divisibility and congruence properties of σ (m)(n), see the surveys in Mitrinovic-S´ andor´ [218] (pp. 92-94). We will return later on the iteration of s(n) when studying sociable numbers.

~~1.8 Pseudoperfect, weird and harmonic numbers~~

W. Sierpinski [286] called a number pseudoperfect if it was the sum of the some its distinct divisors. E.g. 20 = 1+4+5+10 is pseudoperfect but 8 not. Some authors use the ”semiperfect” terminology (see e.g. A. and E. Zachariou [330]).

~~42 PERFECT NUMBERS~~

Numbers n with the property σ(n)>2n are called abundant (for their properties, see Mitrinovic-S´ andor´ [218], and for a recent survey, see Sandor´ [274]. In [218] and [274] one can find results also on deficient numbers, defined by σ(n)<2n). A number is called primitive abundant if it is abundant, but all its proper divisors are deficient; and primitive pseudoperfect if it is pseudoperfect, but none of its proper divisors are. C. Pomerance [236] called n a harmonic number,ifthe harmonic mean of all the divisors of n is an integer (equivalently, σ(n) divides nd(n); where d(n) is the number of divisors of n). These numbers were considered first by O. Ore [228], and some author call them ”Ore numbers”. Finally, S. J. Benkovski [21] called a number weird,ifitisabundant but not pseudoperfect. Now some results on these numbers. Erdos¨ (see [114], p. 46) has shown that the density of pseudoperfect numbers exists, (1) and says that probably there are integers n which are not pseudoperfect, but for which n = ab, with a abundant and b having many prime factors. (2) H. Abbott (see [114], p. 46) lets l = l(n) (where n ≥ 3) be the least integer for which there are n integers 1 ≤ a1 < a2 < ··· < an = l such that each ai divides s = ai (so that s is pseudoperfect). Then

c1 log log n l(n)>n (c1 > 0constant), (3) for all n ≥ 3; and c2 log log n l(n) 0, constant), (4) for infinitely many n. A. and E. Zachariou [330] note that every multiple of a pseudoperfect number is pseudoperfect. (5) All numbers 2m p with m ≥ 1andp aprime between 2m and 2m+1 are primitive pseudoperfect, but there are such numbers not of this form, e.g. 770. (6) There are infinitely many primitive pseudoperfect numbers that are not harmonic numbers. The smallest odd primitive pseudoperfect number is 945. (7) P. Erdos¨ (see [114], p. 47) can show that the number of odd primitive pseudoperfect numbers is infinite. (8) Benkovski and Erdos¨ [22] showed that the density of weird numbers is positive. (9) Some very large weird numbers were found by S. Kravitz [182]. There are 24 primitive weird numbers less than a million (70, 836, 4030, 5830,...). Nonprimitive weird numbers include numbers of form 70p, where p > 144 is a prime, or 836p with p = 421, 487, 491 or ≥ 577; also 7192 · 31. (10) Benkovski and Erdos¨ [22] posed some open problems on weird numbers: are there infinitely many primitive abundant numbers which are weird? (11)

~~43 CHAPTER 1~~

Is every odd abundant number pseudoperfect (thus not weird)? (12) Can σ(n) be arbitrarily large for weird n?(13) They conjecture ”no”. For primitive weird numbers, see also [230]. Clearly, a pseudoperfect number is a perfect one, too. The similar statement for harmonic numbers is due to Ore [228]: n perfect ⇒ n harmonic. (14) Ore proved that a harmonic number has at least two distinct prime factors, (15) and Pomerance [236] (see also D. Callan [41]) proved that the only harmonic numbers with two distinct prime factors are the even perfect numbers. (16) M. Garcia [105] extended the list of harmonic numbers to include all 45 which are < 107, and he found more than 200 larger ones. The least one, apart from 1 and the perfect numbers, is 140. (17) All 130 harmonic numbers up to 2 · 109 are listed in G. L. Cohen [59], (18) and R. M. Sorli (see Cohen and Sorli [66]) has continued the list to 1010.(19) Kanold [170] has shown that the density of harmonic numbers is 0. (20) Ore [228] conjectured that every harmonic number is even, but this probably is very difﬁcult. Indeed, this result, if true, would imply that there are no odd perfect numbers. See also W. H. Mills [215], who proved that if there exists an odd harmonic number n, then n has a prime-power factor greater than 107. Related to Pomerance’s result (16) one can ask: if a harmonic number has three distinct divisors, is it even? (21) In 1998 G. L. Cohen and D. Moujie [62] have introduced a generalization of harmonic numbers. A number n > 1iscalled k-harmonic if nkd(n) Hk = (22) σk(n) k is an integer. Here k ≥ 1 denotes a positive integer, and σk(n) = d denotes d/n the sum of kth powers of divisors of n. The number n is called power-harmonic if it is k-harmonic for some k ≥ 1. A power-harmonic number is proper,itifisk- harmonic for some k ≥ 2. No k-harmonic number with k ≥ 2isknown. However, Cohen and Moujie are able to prove some necessary conditions. For example, if n is power-harmonic, then has at least two distinct prime factors. (23) This generalizes Ore’s result (15). If n is k-harmonic, then

~~Hk(n) ≤ d(n) − 1 (24)~~

For even k-harmonic numbers the following is true: Let n = 2a pb (p odd prime) be a proper k-harmonic number. If k is even, then b ≡ 7 (mod 8);ifk is odd then b is odd and (p + 1)(b + 1) ≡ 0 (mod 16)(25)

~~44 PERFECT NUMBERS~~

~~If n is a proper k-harmonic number, then k − 1 d(n)>1 + · 2k,(26) k + 1 and if n is a proper power-harmonic number, then~~

~~d(n) ≥ 60 (27)~~

~~Also, if n is a proper power-harmonic number, then~~

~~n > 1010 (28)~~

Finally, we note that W. Butske et al. [40] have considered primary pseudoperfect number n which is a product of distinct primes and which satisﬁes 1 1 + = 1. The ﬁrst few such numbers are 2, 6, 42, 1806, 47058,... n p/n p

1.9 Unitary, bi-unitary, inﬁnitary-perfect and related numbers n Adivisor d of n is called a unitary divisor, if d, = 1. Let σ ∗(n) denote d the sum of unitary divisors of n.Itisknown that (see E. Cohen [53], and see also [279] for various divisibility notions, convolutions and arithmetic functions) σ ∗ is multiplicative and σ ∗(pα) = pα + 1 for any prime power α. Also σ ∗(1) = 1. In 1966 M. V. Subbarao and L. J. Warren [303] introduced the unitary perfect numbers n satisfying σ ∗(n) = 2n (1) They found the ﬁrst four unitary perfect numbers, namely

~~6, 60, 90, 87360,(2) and in 1975 Ch. Wall [319], [321] discovered the ﬁfth, namely~~

~~218 · 3 · 54 · 7 · 11 · 13 · 19 · 37 · 79 · 109 · 157 · 313 (3)~~

Subbarao and Warren proved that there are no odd solutions of (1). This was shown independently also by K. Nageswara Rao [224]. No other unitary perfect numbers than the ﬁve given by (2) and (3) are known, but if n = 2am (m odd ) would be a new one, then Subbarao et al. [302] showed that one must have a > 10 and ω(m)>6 (4)

~~45 CHAPTER 1~~

~~Later, Wall [323] improved this to~~

~~ω(m)>8,(5) while in [322] he proved that any new unitary perfect number has a prime power unitary divisor larger than 215 (6)~~

= m a1 ... ar H. A. M. Frei [104] has shown that if n 2 p1 pr is unitary perfect with (n, 3) = 1, then m > 144, r > 144, and n > 10440 (6) It is not known if there are infinitely many unitary perfect numbers. Subbarao (see [300]) has conjectured that there are only finitely many. It is also not known whether there exists at least a unitary perfect number not divisible by 3. (7) (see [303]), or a k-unitary perfect number for k ≥ 3, where σ ∗(n) = kn, (8) see [289]. P. Hagis [124] proved that if (8) holds and n contains t distinct odd divisors, then k = 4or6implies n > 10110, t ≥ 51 and 249|n.(9) If k ≥ 8, then n > 10663 and t ≥ 247, (10) while k ≥ 5, odd, imply n > 10461 and t ≥ 166, 2166|n.(11) In 1989 S. W. Graham [111] proved the following result: If n = 2am is a unitary perfect number, where m is an odd squarefree number, then either a = 1 and m = 3, a = 2 and m = 3 · 5, or a = 6 and m = 3 · 5 · 7 · 13 (thus there are only three such numbers). (12) J. De Boer [129] proved that if n = 29 · 32m is unitary perfect, with (m, 6) = 1 and squarefree, then a = 1 and m = 5. On unitary abundant numbers, see [274]. Adivisor d of n is called a bi-unitary divisor if the greatest common unitary n divisor of d and is 1. Let σ ∗∗(n) be the sum of bi-unitary divisors of n.Wall [320] d call n bi-unitary perfect,if σ ∗∗(n) = 2n (13) It is not difficult to verify that σ ∗∗ is multiplicative and

~~pα+1 − 1 σ ∗∗(pα) = σ(pα) = if α is odd, p − 1 and pα+1 − 1 σ ∗∗(pα) = − pα/2, if α is even (14) p − 1~~

~~46 PERFECT NUMBERS~~

Hence σ ∗∗(n) ≤ σ(n)(15) with equality only if every prime which divides n does so an odd number of times. Wall proves that there are only three bi-unitary perfect numbers, namely 6, 60, 90. (16) Hagis [126] introduced bi-unitary multiply perfect numbers by

n|σ ∗∗(n)(13) and proved that there are no odd numbers of this type. There are 17 numbers n < 107 satisfying (13’) (three of them are the numbers from (16)). W. E. Beck and R. M. Najar [13] essentially have determined all unitary quasiperfect numbers satisfying

~~σ ∗(n) = 2n + 1 (17) as well as all unitary almost perfect numbers such that~~

~~σ ∗(n) = 2n − 1 (18)~~

They proved that there are no unitary quasiperfect numbers, (19) and that n = 1 and n = 2are the all possible unitary almost perfect numbers. (20) A. Bege [16] has proved that there are no bi-unitary quasiperfect numbers, i.e. numbers n satisfying σ ∗∗(n) = 2n + 1. J. Sandor´ [270] (see pp. 11-12) showed there are no unitary quasi superperfect numbers n satisfying σ ∗(σ ∗(n)) = 2n + 1 (21) and that the only unitary almost superperfect number n, i.e. such that

~~σ ∗(σ ∗(n)) = 2n − 1 (22) are n = 1andn = 3. Sitaramaiah and Subbarao [290] rediscovered this result. They study more pro- foundly the unitary superperfect numbers n with~~

~~σ ∗(σ ∗(n)) = 2n (23)~~

The ﬁrst ten such numbers are 2, 9, 165, 238, 1640, 4320, 10250, 10824, 13500 and 23760. There are not known odd unitary superperfect numbers other than 9 and 165, and Sitaramaiah and Subbarao conjecture that the set of such numbers is ﬁnite. (24)

~~47 CHAPTER 1~~

In fact 165 is the single such number, having three distinct prime factors. (25) 238 is the only even unitary superperfect number such that n has three prime factors, while σ ∗(n) two. (26) If n is an odd unitary superperfect number not divisible by 3, then 3|σ ∗(n);in such a case n must be squarefree, ω(n) even, and ω(n) ≥ 52. (27) We quote one more result: All n ≤ 108 such that n|σ ∗(σ ∗(n)) (28) (which could be called as ”unitary multiply superperfect numbers”) are determined; there are 29 such numbers. S. Ligh and Ch. R. Wall [190] have introduced ”nonunitary divisors” and nonunitary perfect numbers. d is a nonunitary divisor of n,ifn = cd, with (c, d)>1. Let σ #(n) be the sum of nonunitary divisors of n.Itisimmediate that

~~σ #(n) = σ(n) − σ ∗(n), (29) and that σ # is not multiplicative. A number n is called nonunitary perfect, if~~

~~σ #(n) = n (30)~~

All known nonunitary perfect numbers are even. P. Hagis, Jr. [129] prove that σ #(n) is odd iff n = 2α M2,whereM > 1isodd and α ≥ 0. (31) As a corollary we get that an odd nonunitary perfect number must be a perfect square. (32) Another result by Hagis says that an odd nonunitary perfect number n must have at least four distinct prime factors, and n > 1015.If3 n,thenω(n) ≥ 7. (33) If 2 · 5 · 7|n,then for such numbers one has also 32||n and 54 · 74|n. Also p n if 11 ≤ p ≤ 271. (34) G. L. Cohen [58] calls a unitary divisor d of n as 1-ary divisor,and he calls d a k-ary divisor of n (fork > 1), and writes d|kn,ifthe greatest (k −1)-ary divisor of d n n and is 1. Note that d, = 1for 1-ary divisors d.Healso calls px an inﬁnitary d d y x y x y divisor of p (y > 0, p prime) if p |y−1 p , and writes p |∞ p .(35) r = y j ,..., Let d be a divisor of n and write n p j , for distinct primes p1 pr , and j=1 r = x j ≤ ≤ = , d p j (where 0 x j y j , j 1 r). Then d is an inﬁnitary divisor of n if j=1

~~x j | y j = , .() p j ∞ p j for each k 1 r 36~~

~~48 PERFECT NUMBERS~~

= j It is interesting to note that, by writing the binary representations x x j 2 j and y − x = z j 2 (where the sums are ﬁnite, j ≥ 0, each x j and z j is0or1,and trailing zeros are allowed where required), then x y p |∞ p iff x j z j = 0 (37)

Another interesting characterization (see [58]) is the following: x y y p |∞ p iff is odd,(38) x y where = C x denotes a binomial coefficient. x y Let σ∞(n) be the sum of all infinitary divisors of n. Then σ∞ is a multiplicative function, and α 2 j j σ∞(p ) = (1 + p ), where y = y j 2 (39) y j =1 The number n is called infinitary perfect,resp. infinitary k-perfect if

σ∞(n) = 2n,(40) resp. σ∞(n) = kn (k ≥ 2)(41) There are no odd infinitary perfect or k-perfect numbers, and Cohen conjectures that there are no infinitary perfect or k-perfect numbers not divisible by 3. (42) Cohen gives a number of 14 infinitary perfect, 13 infinitary 3-perfect, 7 infinitary 4-perfect, 2 infinitary 5-perfect numbers. Pedersen [231] found 139 new infinitary (43) perfect numbers. Cohen proves also that the only infinitary perfect numbers not divisible by 8 are 6, 60 and 90. (44) We note that D. Suryanarayana [306] and K. Alladi [6] have given different generalizations from above for unitary divisors, thus obtaining other notions of k-ary divisors. (45) A number n is called unitary harmonic if σ ∗(n)|nd∗(n) (46) i.e. if the unitary harmonic mean H ∗(n) = nd∗(n)/σ ∗(n) is an integer. The main properties of unitary harmonic numbers were studied by Hagis and Lord [135], but it was earlier introduced also by N. Nageswara Rao [224] who showed that if n is unitary perfect, then it is also unitary harmonic. (47) Ch. R. Wall [Unitary harmonic numbers, Fib. Quart. 21(1983), 18-25] has proved that there are 23 unitary harmonic numbers n with ω(n) ≤ 4, and 43 unitary harmonic numbers n ≤ 106.

~~49 CHAPTER 1~~

Let H ∗(x) be the counting function of unitary harmonic numbers. Then for ε>0 and large x one has H ∗(x)<2.2x1/2 · 2(1+ε) log x/ log log x (48) aresult due to Hagis and Lord [135]. Hagis and Cohen [131] study infinitary harmonic numbers, i.e. those numbers n for which σ∞(n)|nd∞(n)(49) where d∞(n) and σ∞(n) denote the number, resp., sum of infinitary divisors of n. Let H∞(n) = nd∞(n)/σ∞(n). They show that if n is infinitary perfect, then it is also infinitary harmonic. (50) If Sc is the set of all positive integers such that H∞(n) = c, then Sc is finite (or empty) for every real number c.(51) For the counting function H∞(x) of infinitary harmonic function exactly the same estimate as (48) holds:

~~1/2 (1+ε) log x log log x H∞(x)<2.2x · 2 (52)~~

~~1.10 Hyperperfect, exponentially perfect, integer-perfect and γ -perfect numbers~~

In 1975 D. Minoli and R. Bear [217] have generalized the concept of perfect number, by introducing the hyperperfect numbers. An integer n > 1isk-hyperperfect if n = 1 + k[σ(n) − n − 1] (1) When k = 1, this gives the perfect numbers. It easily follows that for k ≥ 2 all k-hyperperfect numbers are odd. For example 21, 2133, 19521 are 2-hyperperfect, 325 is 3-hyperperfect, 301 and 16513 are 6- hyperperfect, 697 is 12-hyperperfect (see [217]). (2) Minoli [216] gave a table for all hyperperfect numbers less than 1500000. All these numbers have two distinct prime factors. But hyperperfect numbers can have more than two prime factors, for example (see te Riele [254]) 13·269·449 = 1570153 is a 5-hyperperfect number, having three distinct prime factors. (3) Minoli and Bear prove that a hyperperfect number cannot be a power of a prime. (4) If n = pq with p, q primes is k-hyperperfect, then k < p < 2k < q.(5) If n = 3m(3m+1 − 2), where 3m+1 − 2isaprime, then n is 2-hyperperfect. (6)

~~50 PERFECT NUMBERS~~

Recently, J. C. M. Nash [226] extended this to any prime p in place of 3, as follows: If n = pm(pm+1 − (p − 1)) and pm+1 − (p − 1) is prime, then n is (p − 1)- hyperperfect. (7) α1 2 Minoli [216] proves that if n = p p is k-hyperperfect, then k + 2 ≤ p1 ≤ (8) 1 2 α α ( + )2 , = 1 2 k 1 (p1 p2 distinct primes); and if n p1 p2 is k-hyperperfect, then α ( 1 + ···+ ) + ( − )( − ) log[k p1 1 p2 1 k 1 ] α2 ≤ (9) log p2 Finally, a conjecture by Minoli and Bear [217] states that there are k-hyperperfect numbers for every k.(10) Another conjecture by them afﬁrms that the converse of theorem (6) is true, namely if n = 3m(3m+1 − 2) is 2-hyperperfect, then 3m+1 − 2isprime. (11) 3k + 1 If k > 1isanodd integer, and p = , q = 3k + 4 are prime, then it 2 is immediate that p2q is k-hyperperfect. J. S. McCranie [69] conjectures that all k- hyperperfect numbers for odd k > 1are of this form. McCranie has tabulated all hyperperfect numbers less than 1011. For unitary hyperperfect numbers, see [122]. Here one replaces in (1) σ(n) with σ ∗(n).Itiseasy to see that if n is unitary hyperperfect, and 3 n, then n ≡ 1 (mod 3). Also, if 3 nk, then n has an odd number of distinct prime factors. n cannot be a prime power, and if n has the form n = paqb, then pa − k = 1, 2, 5, 10, 13, 17, 25, 26, 29, 34,... There are in total 36 unitary hyperfect numbers n ≤ 106.Four of these numbers are unitary perfect, found by Subbarao and Warren, twenty are hyperperfect numbers given by Minoli. From the twelve new ones we quote n = 23 · 5 (k = 3),22 · 13 (k = 3),25 · 32 (k = 7), 32 · 73 (k = 8),27 · 17 (k = 15),etc. D. A. Buell [38] computed all unitary hyperperfect numbers < 108.Bydevel- oping a search procedure different from that of Buell, P. Hagis [127] found all such numbers < 109. α α = 1 ... r If n p1 pr , then E. G. Straus and M. V. Subbarao [299] call d an exponential divisor if d|n and

~~= b1 ... br | ( = , ). ( ) d p1 pr where bi qi i 1 r 12~~

Let σe(n) be the sum of exponential divisors of n. Forvarious arithmetic functions and convolutions on exponential divisors, see Sandor-Bege´ [279]. Straus and Subbarao call n exponentially perfect (or shortly e- perfect)if σe(n) = 2n (13)

~~51 CHAPTER 1~~

Some examples of e-perfect numbers are 22 · 32,22 · 33 · 52,23 · 32 · 52,24 · 32 · 112, 24 · 33 · 52 · 112,26 · 32 · 72 · 132,27 · 32 · 52 · 72 · 132,28 · 32 · 52 · 72 · 1392, 219 · 32 · 52 · 72 · 112 · 132 · 192 · 372 · 792 · 1092 · 1572 · 3132. If m is squarefree, then σe(m) = m,soifn is e-perfect and m = squarefree with (m, n) = 1, then m · n is e-perfect. (14) So it suffices to consider only powerful (i.e. no prime occurs to the first power) e-perfect numbers. Straus and Subbarao proved that there are no odd e-perfect numbers, (15) and that for each r the number of e-perfect numbers with r prime factors is finite. (16) Is there an e-perfect number which is not divisible by 3? (17) Straus and Subbarao conjecture that there is only a finite number of e-perfect numbers not divisible by any given prime p.(18) J. Fabrykowski and Subbarao [100] proved that any e-perfect number not divisible by 3 must be divisible by 2117, greater than 10664, and having at least 118 distinct prime factors. (19) In 1988 P. Hagis, Jr. [128] showed that the density of e-perfect numbers is positive (namely, is 0.0087). (20) The number n is called e-multiperfect (see [299] if

~~σe(n) = kn (21) for some k > 2. Results (15) and (16) are true also for such numbers. W. Aiello, G. E. Hardy and Subbarao [2] proved that if n is e-multiperfect, then~~

~~n > 2 · 107 for k = 3; (22)~~

n > 1085 for k = 4; (23) n > 10320 for k = 5; (24) n > 101210 for k = 6.(25) For the number of powerful solutions n ≤ x of (21), L. Lucht [201] proved that this is ≤ exp(c log x/ log log x)(26) for x ≥ 3(even, for rational k > 1), with c a constant not depending on k. A number of interesting properties are proved in the paper by J. Hanuman- thachari, V. V. Subrahmanya Sastri and V. Srinivasan [139]: If m is squarefree, then m2 is e-perfect iff m = 6. (27) The only e-perfect number having two distinct prime factors is 36. (28)

~~52 PERFECT NUMBERS~~

= bi c j If n is an e-perfect number not multiple of 3, and n pi q j , where ≡ ( ) ≡ ( ) , = 2 3 , p 1 mod 3 , q j 2 mod 3 ,(pi q j primes) with bi Pi Qi Ri (Pi Qi be- ( , ) = = 2 3 , ing squarefree, Pi Qi 1) and e j L j M j N j (L j M j being squarefree and (L j , M j ) = 1), then each Qi = 1, each L j is odd, and no M j is greater than 2. (28 ) = c j = ω( ) = Further, if f the number of q j for which M j 1 and L j is even; g the c j = ω( ) = bi number of q j for which M j 2 and L j is odd, and h the number of pi for which ω(Pi ) is odd, then f + g + h is odd or even, according as

n ≡ 1or2 (mod 3)(29) = a bi c j For numbers divisible by 3 the following is proved: if n 3 pi q j (a > 1, bi > 1, c j > 1), with pi , q j primes, pi ≡ 1 (mod 3), q j ≡ 2 (mod 3) is e-perfect, then either (i) some bi has in its factorization some prime highest to the index congruent to 2 (mod 3) or (ii) some c j has 2 in its factorization highest to the index congruent to 1 (mod 3) or some c j has an odd prime occurring in its factorization to the highest index congruent to 2 (mod 3). Hanumanthachari, Subrahmanya Sastri and Srinivasan study also e-superperfect numbers given by σe(σe(n)) = 2n (30) If n = pq with p, q primes, is a solution of (30), then n = 32.(31) If p is a prime, n squarefree and (p, n) = 1, (p + 1, n)>1, then p2n is e- superperfect only if p = 2. (31) Generally, if (σe(m), n) = 1 and n is squarefree, then mn is e-superperfect iff m is superperfect, too. (32) σ ( 2) σ (4+r)( 2) = · · Related to the iteration of e 3 , they remark that e 3 2 3 5 for any r ≥ 0andconjecture that

σ (k)( 2 · 2) .() e 2 3 is never squarefree 33 Another conjecture states that for every prime p there is a least positive integer k = k(p) such that σ (k)( 2) .() e p is squarefree 34 In this section we shall deal also with integer-perfect numbers,which are numbers n such that = α( ) ( ) n ( ) d d 35 n where indicates that the sum is over the proper divisors d of n and α(d) ∈ (n) {1, −1} for each d. This notion is due to L. V. Marijo [209].

~~53 CHAPTER 1~~

Let ν(n) be the number of such representation of n, and call an integer-perfect number minimal if it has no proper divisor which is integer perfect. Marijo proves the following results: No integer of the form 2ab (b odd) is integer-perfect if b is a perfect square, but otherwise all integers of this form are interger-perfect for sufﬁciently large a.(36) If n is an (ordinary) even perfect number, then

~~ν(n) = ν(2n) = 1.(37)~~

~~Forany positive integer N, there exists a minimal integer-perfect number g(N) such that ν(g(N)) ≥ N.(38) If p is a prime not dividing n, then for any positive integer a,~~

~~ν(pan) ≥ 2a(ν(n))a+1, with equality whenever p >σ(n)(39)~~

~~The following asymptotic results appears in Marijo [210]:~~

card{m ≤ x : ν(m) ≥ m}=O(x3/4). (40) ν(n) For each k ∈ N∗, can be arbitrarily large. (41) nk Finally, we deﬁne γ -perfect numbers.M.D.Miller [214] deﬁnes a function γ : N → N recursively by γ(n) = γ(d)(42) d/n,d 1 will be called γ -perfect if

~~γ(n) = n (44)~~

Miller (who use another terminology) proves that there are no odd γ -perfect numbers. (45) pnq is γ -perfect iff p = 2 and n + 2 = 2q,while pnq2 or pnq3 cannot be γ -perfect. A number of the form pnqr is γ -perfect iff p = 2, and

~~2qr = n2 + 6n + 6.(46)~~

~~54 PERFECT NUMBERS~~

These give example of such perfect numbers, e.g. 48, 1280, 2496, 28672, 454656, 2342912,... More generally, if f is an arithmetic function, then n is called f -perfect by J. L. Pe [On a generalization of perfect numbers, J. Recr. Math. (to appear)] if n = f (d) d|n,d

Recently, Ch. Ashbacher [On numbers that are pseudo Smarandache and Smaran- dache Perfect, Smarandache Notion J. 14(2004), 40-42] has raised the problem of m(m + 1) Z-perfect and S-perfect numbers (where Z(n) = min m ∈ N : n| and 2 S(n) = min{m ∈ N : n|m!}). For n ≤ 106, the only Z-perfect numbers are n = 4, 6, 471544; and the only S-perfect number is n = 12.

~~1.11 Multiplicatively perfect numbers~~

In 2001, Sandor´ [275] has considered the multiplicatively perfect, superperfect, multiperfect etc. numbers. Let T (n) denote the product of all divisors of n > 1. Then n is multiplicatively perfect (in short: m-perfect), if

~~T (n) = n2 (1) m-superperfect,if T (T (n)) = n2 (2) m-multiperfect,if T (n) = nk (3) for some k ≥ 2. He considered also the generalization of (2) to~~

~~T (T (n)) = nk (4)~~

~~(i.e. m-multisuperperfect numbers); as well as~~

~~T (k)(n) = nk (5)~~

The following results are proved: All m-perfect numbers (i.e. satisfying equation = = 3 , (1)) have one of the following forms: n p1 p2 or n p1, where p1 p2 are distinct primes. (6) (The author discovered later that this result appears also in K. Ireland and M. Rosen [154]. However, the ﬁrst source is E. Lionnet [192] who deﬁned ”perfect number of

55 CHAPTER 1 the second kind” as number equal to the product of its aliquot parts. See also [84], p. 58.) There are no m-superperfect numbers. (7) n = 6isthe only perfect number, which is multiplicatively perfect, too. (8) The general form of m-multiperfect numbers can be determined too, e.g.: = 2, = 5 = ; ( ) n p1 p2 or n p1 for k 3 9 = 3 = = 7 = ; ( ) n p1 p2 or n p1 p2 p3 or n p1 for k 4 10 = 4 = 4 = 9 = ; ( ) n p1 p2 or n p1 p2 or n p1 for k 5 11 = 2 = 5, = 11 = ; ( ) n p1 p2 p3 or n p1 p2 n p1 for k 6 12 = 6 = 13 = ; ( ) n p1 p2 or n p1 for k 7 13 = = 3 = 3 3 = 15 = ; ( ) n p1 p2 p3 p4 or n p1 p2 p3 or n p1 p2 or n p1 for k 8 14 = 2 2, = 8, = 17 = ; ( ) n p1 p2 p3 n p1 p2 n p1 for k 9 15 = 4, = 9, = 10 = .() n p1 p2 p3 n p1 p2 n p1 for k 10 16 As corollaries, one can deduce the following assertions: n = 28 is the single perfect number which is m-triperfect, too. (17) There are no perfect and 4-m-perfect numbers; (18) n = 496 is the only perfect number which is 5-m-perfect. (19) Related to equation (4) one can say, that it has at most a ﬁnite number of solution. (20) Particularly, it is not solvable for k = 4, 5, 6. (21) = = 2 = = 3 = For k 3, the general solutions are n 81 ,fork 7, n p1, and for k 9, n = p1 p2 (p1, p2 distinct primes). (22) Equation (5) has no solutions n > 1. (23) The function T (n) has some interesting properties: Forexample log T (n) log T (n) lim inf = 1, lim sup = +∞; (24) →∞ n log n n→∞ log n the normal order of magnitude of

~~log log T (n) is (1 + log 2) log log n − log 2 (25)~~

F. Luca [200] proves that T (σ (n)) > T (n) for almost all n, and that T (n) = T (σ (n)) is impossible for any n > 1. He shows also that both of the relations T (n)|T (σ (n)) and T (σ (n))|T (n) have inﬁnitely many solutions.

~~56 PERFECT NUMBERS~~

T. Salˇ at´ and J. Tomanova´ [267] have shown that log log T (n) → 1 + log 2 as n →∞ log log n on a sequence of density 1. A similar result holds true for T (n)/n. Recently Z. Weiyi [On the divisor product sequences, Smarandache Notions J. 14(2004), 144-146] has obtained the following results: 1 1 = x + C + O , ( ) log log 1 n≤x T n log x 1 log log x = π(x) + (log log x)2 + B log log x + C + O ( ) 2 n≤x T n log x where x ≥ 1, C1, B, C2 are constants, and T (n) = T (n)/n (π(x) is the number of primes ≤ x). A. Bege [17] introduced unitary m-perfect numbers n > 1assolutions of

T ∗(n) = n2 (26) where T ∗(n) is the product of all unitary divisors of n. Similar concepts as above for unitary m-superperfect and unitary multiperfect are also introduced and studied. All unitary m-perfect numbers have two distinct prime factors; (27) there are no unitary m-superperfect numbers. (28) If T ∗(k)(n) = nl (29) (where T ∗(k) denotes the k-times iteraton of T ∗) then n > 1iscalled m-unitary (k, l)-perfect number. If l = 2mk, then there are no m-unitary (k, l)-perfect numbers, (30) while if l = 2mk, then every n with ω(n) = m + 1isanm-unitary (k, l)-perfect number. (31) In a forthcoming paper, A. Bege [18] considered the similar problems for bi- unitary divisors. n > 1ism-bi-unitary perfect,if

T ∗∗(n) = n2,(32) where T ∗∗(n) is the product of bi-unitary divisors of n etc. All m-bi-unitary perfect = 4 = 3 = 2 2 = 2 numbers have one of the following forms: n p1, n p1, n p1 p2, n p1 p2, n = p1 p2, where p1 = p2 are primes. (33)

~~57 CHAPTER 1~~

All m-bi-unitary superperfect numbers have one of the following forms: n = p, n = p2 (p prime). (34) Let k ≥ 2. All (l, k) m-bi-unitary perfect numbers have the form − = ki ki 1 ( ) n pi pi 35 ki =2k ki =2k+1

l where k = ki and pi (i = 1, l) are distinct primes. i=1 In a recent paper, J. Sandor´ [276] has introduced the multiplicatively e-perfect numbers. Let Te(n) denote the product of all e-divisors of n > 1. Then n is called m − e-perfect if 2 Te(n) = n (36) He proved the following theorem: n > 1ism − e-perfect iff n = pk, where p is an arbitrary prime, and k is an arbitrary (ordinary) perfect number (i.e. σ(k) = 2k). (37) Forexample, n = p6, p28, p496,... are all m − e-perfect numbers. Analogously, the m − e-superperfect numbers with the property

2 Te(Te(n)) = n (38) have the form n = ps, where s is an ordinary superperfect number (i.e. σ(σ(s)) = 2s). (39) According to the open problems relating to the ordinary perfect, or superperfect numbers, the complete determination of m − e-perfect numbers is also left to the future.

~~1.12 Practical numbers~~

A positive integer m is said to be practical (see A. K. Srinivasan [293]) if every integer n, with 1 ≤ n ≤ σ(m) is a sum of distinct divisors of m.(1) α α = 1 ... r < < ··· < B. M. Stewart [298] showed that m p1 pr with p1 p2 pr primes and integers αi ≥ 1(i = 1, r)ispractical iff either m = 1orp1 = 2 and for α α α = ,..., ≤ σ( 1 2 ... i−1 ) + every i 2 r, pi p1 p2 pi−1 1. (2) In 1950 P. Erdos¨ [87] announced that practical numbers have asymptotic density. (3) Let 1 = d1 < d2 < ··· < dr = n be the divisors of n and for k ≤ r put tk = d1 + ···+dk with t0 = 0. Then n is practical iff

~~dk+1 ≤ tk + 1 for all k = 0, r − 1.(4)~~

~~58 PERFECT NUMBERS~~

As a corollary, if n is practical, then σ(n) ≥ 2n − 1.(5) Results (4) and (5) are due to D. F. Robinson [261]. Let P(x) be the cardinality of practical numbers n ≤ x. Then 1 P(x) = O(x/(log x)β ) for every β< (1/ log 2 − 1)2 = 0.0979 ... (6) 2 and 1 P(x) ≥ Ax exp (log log x)2 + 3 log log x (7) 2 log 2 for sufficiently large x (where A = 25/2/5). Here (6) is due to M. Hausman and H. N. Shapiro [142], and (7) to M. Margen- stern [208]. Margenstern proved also that if n is practical, distinct from a power of 2, then σ(n) ≥ 2n,(8) which is related to (5). As a corollary of the proof of (8) one gets: all even perfect numbers are practical (see [209]). (9) ≥ / Hausman√ and Shapiro proved also the fact that for all x 1 3, the interval (x, x + 3 x) contains a practical number. (10) Margenstern proved also that lim inf σ(m)/m = 2, lim sup σ(m)/m =∞,(11) →∞ m n→∞ where m is practical, distinct from powers of 2. He conjectured that every even positive integer is a sum of two practical numbers, (12) and that there exist infinitely many practical numbers m such that m − 2andm + 2 are also practical. (13) Conjectures (12) and (13) have been proved affirmatively by G. Melfi [213] (who used also a table by Margenstern for all even number < 100000 for (12)). Finally we state another conjecture (which is still open) by Margenstern, namely that x P(x) ∼ λ (x →∞)(14) log x where λ ≈ 1.341. E. Saias [266] proved that x x c · ≤ P(x) ≤ c · for x ≥ 2 (15) 1 log x 2 log x for all x ≥ 2; while G. Melfi [212] showed that there exist infinitely many practical numbers in the sequences of Fibonacci, Pell, and Lucas. (16)

~~59 CHAPTER 1~~

~~1.13 Amicable numbers~~

~~Two numbers a and b are called amicable if~~

~~σ(a) = σ(b) = a + b~~

Clearly for a = b we reobtain the perfect numbers, so one may assume in what follows that a = b.According to Iamblicus (about 283-330), ”certain men steeped in mistaken opinion thought that the perfect numbers was called love by Pythagoreans on account of the union of different elements and affinity which exists in it; for they call certain other numbers, on the contrary, amicable numbers, adopting virtues and social qualities to numbers, as 284 and 220, for the parts of each have the power to generate the other, according to the rule of friendship, as Pythagoras affirmed. When asked what is a friend, he replied ”another I”, which is shown in these numbers. Aristotle so defined a friend in his Ethics.” Among Jacob’s presents to Esau were 200 she-goats and 20 he-goats, 200 ewes and 20 rams (Genesis, XXXII, 14). Abraham Azulai (1570-1643), in commenting on this passage from the Bible, remarked that he had found written in the name of Rau Nachson (ninth centure A.D.): ”Our ancestor Jacob prepared his present in a wise way. This number 220 (of goats) is a hidden secret, being one of a pair of numbers such that the parts of it are equal to the other one 284, and conversely. And Jacob had this in mind; this has been tried by the ancients in securing the love of kings and dignatories.” The Arab El Madschrˆıtˆı (1007) of Madrid related that he had himself put to the test the erotic effect of ”giving any one the smaller number 220 to eat, and himself eating the larger number 284”. Ben Kalonymos discussed amicable numbers in 1320 in a work written for Robert of Anjou. A knowledge of amicable numbers was considered necessary by Jochanan Allemanno (fifteenth century) to determine whether an aspect of the planets was friendly or not. In 1634 Mersenne remarked that ”220 and 284 can signify the perfect friendship of two persons since the sum of the aliquot parts of 220 is 284 and conversely, as is these two numbers were only the same thing”. For the early history of amicable numbers (up to 1919), we cite L. E. Dickson [84]. E. B. Escott [95] made a review of the subject up to 1941, while that of E. J. Lee and J. S. Madachy [187] to 1972. More recent surveys can be found in Guy [114], Mitrinovic-S´ andor´ [218], Eric Weisstein [325]. Here we will study the subject in a such a manner which shows that certain for- gotten or new aspects could be important for the further considerations. Only the most important results will be pointed out.

~~60 PERFECT NUMBERS~~

The first method described for finding amicable pairs is due to Thabit ibn Qurra (836-901) (see [84]): Let p = 3 · 2n−1 − 1, q = 3 · 2n − 1, r = 9 · 22n−1 − 1beall primes (where n is a positive integer). Then a = 2n pq and b = 2nr (1) are amicable. For n = 2 one has p = 5, q = 11, r = 71, so one gets the Pythagorean pair. By rediscovering Thabit’s rule, Fermat in 1636 (in a letter for Mersenne) discovered for n = 4 (when p = 23, q = 47, r = 1151) a new pair a = 17296, b = 18416. In 1838 Descartes (again in a letter to Mersenne) found for n = 7(when p = 191, q = 383, r = 73727) the third pair a = 9363584, b = 9437056. It was discovered later that these pairs were known to Ibn-al-Banna(´ 1256-1321) of Marakesh and to K. Farisi of Bagdad (see e.g. W. Borho [29]). Despite later searches to n ≤ 8byEuler, to n ≤ 15 by Legendre, to n ≤ 34 by Le Lasseur, to n ≤ 200 by Gerardin,´ to n ≤ 1000 by Riesel (see [187]), no additional Thabitean pair have been discovered. L. Euler (1747) [96] was the first to treat extensively the subject of amicable numbers and to show for the first time their great diversity. To exhaust all amicable numbers of the form (1) Euler generalized Thabit’s formulae as follows: n−β β If for g = 2 + 1 with some β,0<β

n+β 2 s = (r1 + 1)(r2 + 1) − 1 = 2 g − 1 (2) are prime, then the numbers (1) are amicable, and this gives all amicable pairs of the form (1). For example, β = n − 1gives Thabit’s rule. Euler’s generalization gives further amicable pairs for n = 8, β = 1, resp. n = 40, β = 29, as discovered by A. M. Legendre and P. Chebyshev, resp. by te Riele [252]. For various other algebraic methods due to Euler, see [187]. We note that a simple method, by the use of Diophantine equations was discovered earlier by F. van Schooten in 1657, but his method gives only the ﬁrst three known pairs (see [84]). Euler’s methods enabled him to give a list of 30 new pairs. One of the Euler’s method (the third algebraic method) was discovered independently also by G. W. Kraft (1751), see [84], [187]. In 1968 E. J. Lee [186] discovered the so called ”BDE method” (i.e. Bilinear Diophantine Equation method). Given natural numbers a1, a2 one may determine all amicable pairs of the form

~~a = a1q, b = a2s1s2 (3) where q, resp. s1, s2 (s1 = s2) are primes not dividing a1,resp. a2,bysolving a~~

~~61 CHAPTER 1~~

~~bilinear Diophantine equation on s1, s2 as follows: Take any factorization of the number ( f + d) f + dg = d1d2 (4) into two different natural factors d1, d2, where~~

~~f := (σ (a ) − a )σ (a ), g := a σ(a ), 1 1 2 1 1 (5) d := a2σ(a1) + a1σ(a2) − σ(a2)σ (a1)~~

~~If then, for i = 1, 2, si = (di + f )/d (6) are integers, prime, and prime to a2, and if also~~

−1 q = σ(a2)σ (a1) (s1 + 1)(s2 + 1) − 1 (7) is prime, and prime to a1, then we have an amicable pair (3), and this gives all such pairs. The idea of this method goes essentially back to Euler, who formulated, and extensively used it in several special cases, as did many authors later on. Anew analogue of Thabit’s formula has been discovered in 1972 by W. Borho [28], and later by E. Borho and H. Hoffman [31]. We cite here the later result: Let n be a positive integer, and choose β,0<β

= α + ( n+1 − ) , = n−α − p 2 2 1 g r2 2 gp 1 and ( ) n−α+β 2 9 s = (r1 + 1)(r2 + 1) − 1 = 2 g p − 1 are also prime. Then n n a = 2 pr1r2, b = 2 ps (10) are amicable pairs. Forexample, n = 2andα = β = 1give the amicable numbers a = 22 ·23·5·137, b = 22 · 23 · 827, discovered by other methods by Euler. Anew analogue of Euler’s formula is contained in the following result ([31]): Let n,γ be positive integers with 0 <γ

~~n+1 n−γ p = D + 2 − 1, r1 = pf − 1, r2 = (r1 + 1)2 − 1incase (i) (12)~~

~~62 PERFECT NUMBERS resp. n n n−γ r2 = 2 + f − 1, r2 = p(2 + f )2 − 1incase (ii) (13)~~

~~s = (r1 + 1)(r2 + 1) − 1 (14)~~

~~Whenever the four numbers p1, r1, r2, s are primes, then~~

~~n n a = 2 pr1r2, b = 2 ps (15) are amicable, and all amicable pairs (15) with p, r1, r2, s odd primes, are obtained in this way. Forexample, n = 8, γ = 7, f = 23 · 31 give Lee’s pair [186]~~

~~a = 28 · 1039 · 503 · 1047311, b = 28 · 1039 · 527845247.~~

In 1866 a 16-year old Italian school boy, N. I. Paganini found the small amicable pair (1184, 1210), which is not in the list by Euler (but he gave no indication of the method of discovery). (16) E. B. Escott was the most proliﬁc discoverer of amicable numbers prior to the computer era. There were 390 known amicable pairs as of 1946, (see Escott [95]) and 219 of them were due to Escott. (17) H. L. Rolf [262] was the ﬁrst (by using an exhaustive numerical method) to discover an amicable pair on a computer. He investigated all numbers less than

~~105 (18)~~

~~J. Alanen, O. Ore and J. Stemple [5] extended the numerical search to~~

~~106 (19) discovering 8 new pairs. In 1970 H. Cohen [54] extended the range to~~

~~108.(20)~~

In 1984 te Riele [256] discovered a trick named ”daughter pair from mother pairs”, with a remarkable efﬁciency. This can be summarized as follows: take the in- puts a1, a2 from the list of already known amicable numbers (see the BDE-method (3)-(7)) (a, b) by splitting both numbers a, b into a = a1v1, b = a2v2, where vi (i = 1, 2) is either 1, or a large simple prime factor of a, b (v1 for a, v2 for b). From a list of known (”mother”)-pairs new (”daughter”) pairs can be computed. (21) Foranimprovement of te Riele’s trick, see the ”breeding” method by Borho and Hoffman [31].

~~63 CHAPTER 1~~

te Riele [257] has found all 1427 amicable pair whose lesser members are less than 1010 (22) Moews and Moews found 3340 numbers less than 1011 (D. Moews, P. C. Moews [221]) and 5001 numbers less than 3 · 06 · 1011 (D. Moews, P. C. Moews [222]). In 1993 [258] te Riele discovered a new method similar to the Thabit-Euler-Lee- Borho-Hoffman methods. In 1997 M. Garcia (see [325]) discovered a new amicable pair, each of whose members has 4829 digits. Probably this is momentary the largest known amicable pair. (23) The new pair has the form

~~a = CM[(P + Q)P89 − 1] (24) b = CQ[(P − M)P89 − 1] where C = 211 · P89,~~

~~M = 287155430510003638403359267,~~

P = 574451143340278962374313859, Q = 136272576607912041393307632916794623. Note that P, Q,(P + Q)P89 − 1and(P − M)P89 − 1are all primes. During the first three months of the year 2001, M. Garcia has found over one million new amicable pairs (see his article ”A million new amicable pairs” from J. Integer Sequences (electronic) 4(2001), article 01.2.6). The earliest known amicable numbers all were divisible by 3. So P. Bratley and J. McKay [32] conjectured that there are no amicable pairs coprime to 6. However, S. Battiato and W. Borho [11] found a counterexample. (25) Today there are many known amicable pairs not divisible by 6 (see J. M. Pedersen [232]). (26) The first example of a pair coprime to 30 has been found by Y. Kohmoto (see [232]) in 1997, consisting of a pair of numbers each having 193 digits. (27) No amicable pairs which are coprime to 210 are currently known. (28) It is also not known, if there exist odd amicable pairs with one member, but not both divisible by 3. (29) A conjecture by Ch. Wall says that odd amicable pairs must be incongruent (modulo 4) (30) (see Guy [114], p. 57). It is not known whether or not there are an infinite number of amicable pairs. (31)

~~64 PERFECT NUMBERS~~

The ﬁrst result in this direction is Kanold’s proof [166] in 1954 that the density of amicable pairs is ≤ 0, 204 (32) The fact that this density is zero, is due to P. Erdos¨ [88]. Formulated more precisely, let

~~B(x) = card{(a, b) : a < b, a < x,(a, b) amicable pair}~~

~~Then B(x) = o(x)(33) Erdos¨ and G. J. Rieger [94] improved this to~~

~~B(x) = O(x/ log log log x)(34)~~

Finally, in 1981 C. Pomerance [241] deduced the strongest known estimates, namely B(x) ≤ x exp(−(log x)1/3)(35) resp. B(x) x exp(−c(log x log log x)1/3)(36) Note that (35) has an interesting Corollary, not known before, namely that the sum of the reciprocals of the amicable numbers is ﬁnite. (37) Relatively prime amicable pairs are not known, (38) but conditions on their existence have been studied by O. Gmelin [109], H.-J. Kanold [163] and P. Hagis, Jr. [115], [116]. Hagis showed that there are no relatively prime amicable numbers with lesser member less than

~~1060 (39)~~

~~Gmelin proved that an even-odd amicable pair (a, b) must be of the form~~

~~a = 2α A2, b = B2,(A, B odd)(40)~~

~~A. A. Gioia and A. M. Vaidya [108] show that A in (40) cannot be a square. (41) If in (40) α>1, then A cannot be a prime~~

~~≡ 4 (mod 3)(42)~~

~~B must be composite, and if α is odd, then (A, 3) = (B, 3) and there exists a prime q and a positive integer γ such that~~

~~qγ ||B and q ≡ γ ≡ 1 (mod 3). (43)~~

~~65 CHAPTER 1~~

If α ≡ 3 (mod 4), then there exists a prime p and a positive integer δ such that pδ||B and α + 1 2p ≡ δ ≡ 2 (mod 5), and A ≡ B ≡ σ(A2) ≡ 0 (mod 5). (44) 4 (42)-(44) are due to Gioia and Viadya, too. Kanold [164] proves that if α>1in(40), then a < b.(45) If α = 1 and a > b in (40), then b must contain at least ﬁve distinct prime factors. (46) Also, if a, b in (40) are relatively prime, then

~~ab ≡ 2 (mod 24)(47)~~

Acorollary of (47) is the interesting fact that there are no relatively prime even- odd amicable pairs in which both members are perfect squares. (48) If A in (40) is a pure prime power, then α = 1, and if we put M = pk, then k > 6, ω(B)>24, p ≡ 1 (mod 12), and a > b > 1075.(49) Another result by Kanold states that if ω(a) = ω(b) = 2, then a and b cannot be an amicable pair. (50) Lee and Madachy [187] prove that if a < b are amicable numbers with a and b even, then a 1 > (51) b 2 Borho [30] proved that the set of relatively prime amicable pairs a, b with ω(a)+ ω(b) ≤ K (where K is a given bound), is ﬁnite. (52) P. Hagis, Jr. [117] obtained the following results: Let a and b be relatively prime amicable pairs of opposite parity such that a < b and b even. Put 1 S = (p prime) p|ab p If 5|ab, then S < 1.57549, (53) while if 5 ab, then S < 1.59862. (54) These bound are true also, if b < a,buta < 2b.(55) If a, b are relatively prime amicables with opposite parity, then

~~S > 1.43151 if 5|ab; (56)~~

~~S > 1.45382 if 5 ab.(57) In 1986 te Riele [257] found 37 pairs of amicable pairs (a, b) and (c, d) such that~~

~~a + b = c + d (58)~~

~~66 PERFECT NUMBERS~~

The first such pair is (609928,686072) and (643336,652664). Three such pairs were obtained in 1993 by Moews and Moews, while te Riele in 1995 obtained four pairs. In 1997 five, resp. six pairs have been discovered. Related to amicable numbers in special sequences or functions, we quote the following results: For any a, the numbers a and ϕ(a) (ϕ is Euler’s totient) cannot be amicable. (59) The same for a and (2b − 1)a + 1, where a, b > 1. (60) Results (59) and (60) are due to F. Luca [196]. Here an open question is stated too: for what value of a are a and a + 1 amicable? (they cannot be both perfect; see 4.(49)). (61) The Pell sequence (Pn) is given by P0 = 0, P1 = 1, Pn+2 = 2Pn+1 + Pn for n ≥ 0. In [199] it is proved that there are no amicable Pell pairs. (62) The notion of amicable numbers (or pairs) has been studied also in certain other contexts. Unitary amicable numbers have been introduced by Hagis [118], and studied by him, and later by Garcia [106]. It seems that there are more than 100 such pairs. The first few are (114,126), (1140,1260), (18018,22302),... The largest known unitary amicable pair has its members with 192 digits, discovered by Y. Kohmoto [327]. We quote some theoretical results on unitary amicable pairs (a, b), i.e. satisfying σ ∗(a) = σ ∗(b) = a + b.For example, neither a nor b is of the form pα (prime power); they are of the same parity; if a = 2α M, b = 2β N, where β>α>0 and M, N are odd integers having s, resp. t prime divisors, then s ≤ α and t ≤ α.(63) If a = 2M, b = 2N, where (10, MN) = 1, then 10a and 10b also is a pair of unitary amicable numbers. If a and b are both squarefree, then (a, b) is unitary amicable pair iff it is an amicable pair. (63’) It is not known if there are an infinite number of unitary amicable pairs, or if there exists a such pair with relatively prime components. For the bi-unitary amicable numbers see another paper by Hagis [126]. Let (a, b) be a bi-unitary amicable pair, i.e. σ ∗∗(a) = σ ∗∗(b) = a + b.Thena and b must have the same parity. If a = 2α M, b = 2β N with M ≡ N ≡ 1 (mod 2), α<β, then ω(M) ≤ α, ω(N) ≤ β.(64) As a corollary one gets that if a = 2M, b = 2β N with β>1 and M, N odd, then M = pc, N = qd (prime powers). Suppose that a = mM, b = mN with (m, M) = (m, N) = 1 and where every exponent in the prime factorization of M and N is odd. If σ ∗∗(n)/n = σ ∗(m)/m and (n, M) = (n, N) = 1, then (nM, nN) is a new bi-unitary amicable pair. (64’) There are sixty pairs of bi-unitary amicable pairs (a, b) with a < b and a ≤ 106. The first examples are (114,126), (594,846), (1140,1260), (3608,3952), (4698,5382),...

~~67 CHAPTER 1~~

~~Let σ #(n) be the sum of nonunitary divisors of n. The numbers a and b are called nonunitary amicable (see [190]) if~~

~~σ #(a) = b and σ #(b) = a.(65)~~

~~All known nonunitary amicable pairs are even. In [129] Hagis proves that if (a, b) is a pair of odd nonunitary numbers, then~~

~~a = M2, b = N 2,(66) while if such a pair is even-odd, then~~

~~a = 2α M2 with M odd and α ≥ 1 (67)~~

~~Garcia (see [114], p. 59) has called a pair of numbers (a, b) quasi-amicable if~~

~~σ(a) = σ(b) = a + b + 1 (68)~~

Forexample, (48,75), (140,195), (1575,1648), (1050,1925) are quasi-amicable pairs. Some authors use the ”betrothed numbers”terminology, but remarking that for a = b one reobtains from (68) the quasi-perfect numbers (see section 6), this one is more natural terminology. Hagis and Lord [136] have found all 46 pairs of quasi-amicable numbers with a < 107, a < b.(69) All of them are of opposite parity. No pairs are known with a, b having the same parity. (70) Hagis and Lord prove that if a < b are of the same parity, then

~~a > 1010.(71)~~

~~If (a, b) = 1isaquasi-amicable pair, then~~

~~ω(ab) ≥ 4,(72) while if (a, b) = 1 with a, b odd, then~~

~~ω(ab) ≥ 21.(73)~~

~~Finally, if (a, b) = 1 and a, b are of the same parity, then~~

~~a and b are > 1030 (74)~~

~~The numbers a and b will be called almost-amicable if~~

~~σ(a) = σ(b) = a + b − 1 (75)~~

~~68 PERFECT NUMBERS~~

Note that for a = b one reobtains the almost-perfect numbers (see section 6). Beck and Najar [12] (who call a, b satisfying (75) ”augmented amicable”, while for (68) they use the terminology ”reduced amicable”) found 11 almost-amicable pairs. (76) They found no unitary quasi or almost amicable pairs (a, b) such that a < b, b < 105,(77) i.e. satisfying (68), resp. (75) with σ ∗ in place of σ. Cohen [58] has introduced infinitary amicable numbers (see section 9 for infinitary divisors) a, b defined by

σ∞(a) = σ∞(b) = a + b (78) Such pairs are e.g. (114,126), (594,846), (1140,1260), (4320,7920), (5940,8460), (8640,11760). He lists all inﬁnitary amicable numbers (a, b) with a < b and a < 106. There are 62 such pairs. (79) From the list it is interesting to remark that very often a and b have similar prime factorization, and Cohen asks for a motivation of this fact. (80) In an unpublished paper, I. Z. Ruzsa [265] called the numbers a and b lovely numbers if σ(a) = 2b,σ(b) = 2a (81) All lovely pairs (a, b) with a and b even are given by a = 2k(2q+1 − 1), b = 2q (2k+1 − 1)(82) where 2q+1 − 1 and 2k+1 − 1 are primes. Sandor´ [277] obtains a new proof of this result. He also proves that if (a, b) is a super-lovely pair, deﬁned by

σ(σ(a)) = 2b,σ(σ(b)) = 2a (83) with a and b even, then a = b,soone reobtains the even superperfect numbers (see section 7). The unitary, etc. analogue of lovely numbers are introduced in [278]. In 1913 Dickson [81] deﬁned amicable triples (a, b, c) by σ(a) = σ(b) = σ(c) = a + b + c (84) After developing a theory analogues to Euler, he obtained eight sets of amicable triplets in which two of the numbers are equal, and two triples of distinct numbers ( · , · , ), 293 337a 5 16561a 993371a resp. (85) (3 · 89b, 11 · 29b, 359b) where a = 25 · 3 · 13, b = 214 · 5 · 19 · 31 · 151.

~~69 CHAPTER 1~~

A. Makowski [204] gave by other method new amicable triples, e.g. (22 · 32 · 5 · 11, 25 · 32 · 7, 22 · 32 · 71). In 1921 T. E. Mason [211] further generalized amicable numbers by introducing amicable k-tuples a1, a2,...,ak where

~~σ(a1) = σ(a2) = ···=σ(ak) = a1 + a2 + ···+ak (86)~~

It is not known if there exist inﬁnitely many such k-tuples (for any k ≥ 2). (87) Y. Kohmoto (see [114], p. 59) found an example of amicable 4-tuples (which he calls quadri-amicable numbers) with all of ai (i = 1, 4) not multiple of 3. (88) The smallest known amicable quadruple is

~~(842448600, 936343800, 999426600, 1110817800).~~

B. F. Yanney [329] defined a k-tuplet of amicable numbers a1,...,ak by a1 + a2 + ···+ak = (k − 1)σ (ai )(i = 1, k).Fork = 2 this definition coincides with that of Dickson. For k > 2, however the two definitions are distinct. E.g. a1 = 308, a2 = 455, a3 = 581 are a triplet of amicable numbers in the sense of Yanney, since σ(a1) = σ(a2) = σ(a3) = 672 and a1 + a2 + a3 = 2 · 672. C. W. Anderson and D. Hickerson [7] call the pair (a, b) a friendly pair,ifa = b and σ(a) σ(b) = (89) a b Forexample, (6n, 28n) for (n, 42) = 1 are such pairs. A number a is called solitary if it doesn’t exist b = a such that (a, b) is a friendly pair (90) (i.e. a is a number ”without friends”). Clearly, if a = b are perfect numbers (i.e. σ(a) = 2a, σ(b) = 2b), then (a, b) satisfies (89). An odd solution is (135,819) or (33 · 5 · 7, 34 · 72 · 112 · 192 · 127); while a solution with a even, b odd is

~~(42, 32 · 5 · 72 · 13 · 19). (91)~~

M. G. Greening [7] shows that numbers n such that (n,σ(n)) = 1, are solitary. (92) There are 53 numbers less than 100, which are solitary, but there are some numbers, as 10, 14, 15, 20,... for which we cannot decide if they are solitary or not. (93) The density of friendly pairs is positive, ﬁrst shown by Erdos¨ [89]: the number of solutions of (89) satisfying a < b ≤ x equals Cx + o(x),where C > 0isaconstant (in fact C ≥ 8/147, see [7]). (94)

~~70 PERFECT NUMBERS~~

If (a, b) = 1isafriendly pair, then (89) implies that a|σ(a), b|σ(b),soa and b are multiply perfect numbers (see section 5). It is not known if (89) has inﬁnitely many solutions satisfying (a, b) = 1. (95) If a and b are squarefree, then (a, b) cannot be a friendly pair (see [89]). (96) The density of numbers a with friends is unity, (97) see [7]. R. D. Carmichael [45] called a pair (a, b) multiply amicable if σ(a) = σ(b) = k(a + b) for some positive integer k.(98) For k = 1weobtain the classical amicable pairs. Mason [211] gives various multiply amicable pairs for k = 2 and k = 3. Cohen, Gretton and Hagis [60] called the integers a and b < a (α, β)-multiamicable if σ(b) − b = αa, σ(a) − a = βb, with α, β positive integers. They proved that b cannot have just one distinct prime factor, and if it has precisely two distinct prime factors, then α = 1andb is even. For small α, β various (α, β)-multiamicable numbers have been tabulated. σ(a) M + αN N + β M Suppose that (a, M) = (a, N) = 1 and = = . Then a σ(M) σ(N) aM, aN are (α, β)-multiamicable. By applying this proposition, the authors have found 72 more multiamicable pairs. They proved also that the density of multiamicable numbers is zero. Cohen and te Riele [65] have studied multiple ϕ-amicable pairs (where ϕ is Euler’s totient). The pair (a, b) with 1 < a ≤ b is called ϕ-amicable pair with multiplier k (denoted as (a, b; k) if

~~a + b ϕ(a) = ϕ(b) = , for some integer k ≥ 1.(99) k~~

~~The pair (a, b) is ϕ-amicable if there exists k with (99). In fact, it is easy to see that one must have k > 2. A ϕ-amicable pair (a, b; k) is called primitive if gcd(a, b) is squarefree and~~

~~gcd(a, b; k) = 1.(100)~~

Cohen and Riele have computed all ϕ-amicable pairs with b < 109, and found 812 pairs for which the greatest common divisor is squarefree. Among them 499 are primitive. In fact with a ϕ-pair for which the g.c.d. is squarefree, inﬁnitely many other ϕ-amicable pairs can be associated. (101) They proved the following general theorems: There are only ﬁnitely many primitive ϕ-amicable pairs with a given number of different prime factors. (102)

~~71 CHAPTER 1~~

~~Let (a, b; k) be ϕ-amicable, where ω(a) = r, ω(b) = s. Let p be the smallest prime divisor of ab, and put m = max{r, s}. Then~~

~~k + 4m − 8 p ≤ with k ≥ 5, if the pair is even, k − 4 (103) k + 2m − 2 p ≤ with k ≥ 3, if the pair is odd. k − 2 If (a, b; k) is a ϕ-amicable pair with (a, b) = 1, then~~

~~ab k2 k − 1 < < (104) ϕ(ab) 4~~

As a corollary of (104) one obtains that if q is a prime, q < a, and (q, a) is ϕ-amicable, then ω(a) ≥ 12.(105) Finally, except for the pair (4,6) if one member of a ϕ-amicable pair has exactly two distinct prime factors, then the other member has at least four distinct prime factors.

~~1.14 Sociable numbers~~

~~In this ﬁnal section we shall treat the iteration properties of special arithmetic functions like s(n) = σ(n) − n (1) By considering the aliquot sequence~~

{s(k)(n)} for k = 0, 1, 2,... (2) where s(0)(n) = n,in1902 A. Cunningham [73] said that the chain n, s(n), s(2)(n),... is of period k if s(k)(n) = n.The numbers in a chain of period k form a cycle (or period k). For k = 1, when s(n) = n one reobtains the perfect numbers, while for k = 2, when s(s(n)) = n, the numbers n and s(n) will be amicable. Even in 1887 E. Catalan [49] stated empirical results (in corrected versions by Dickson [83]) on the aliquot sequence. For example, that every non-periodic chain contains a prime (3) (so the next term is 1, since s(p) = 1 for a prime p), For k odd, k > 1, there is no chain an1, an2,...,ank of period k in which (n1,...,nk) = 1and(n j , a) = 1, j = 1, k, a > 1. (4)

~~72 PERFECT NUMBERS~~

~~In 1918 P. Poulet (see [244]) discovered a chain of period 5, namely~~

~~n = 12496 = 24 · 11 · 71,~~

s(n) = 24 · 19 · 47, s(2)(n) = 24 · 967, s(3)(n) = 23 · 23 · 79, s(4)(n) = 23 · 1783, s(5)(n) = n (5) and another one of period 28 for n = 14316. (6) After a gap of 50 years, and the advent of high-speed computers, H. Cohen (see [114]) discovered nine cycles of period 4, and Borho [27], as well as David and Root (see [114]) also discovered some. (7) Moews and Moews [220] have made an exhaustive search for such cycles with greatest member less than 1010. There are 24 such cycles. (8) They found also an 8-cycle. A. Flammenkamp [101] discovered a new 8-cycle, and one 9-cycle. (9) Moews and Moews [221] have continued to uncover all cycles, of any length, whose member preceding the largest member is less than 3.6 · 1010. They found three more 4-cycles, and one 6-cycle, all of whose members are odd (for n = 21548919483 = 35 · 72 · 13 · 17 · 19 · 431).(10) No one found a 3-cycle, and it has been conjectured that such cycles do not exist. (11) On the other hand it has been conjectured that for each k there are infinitely many k-cycles (see [114]). (12) Fornew four-cycles, see [25] (where fifty new such cycles are found, bringing the total number of 4-cycles to 110). Related to Catalan’s conjecture (3), today we have experimental (and heuristic) arguments that some sequences go to infinity (see e.g. [251]). (13) Perhaps (13) holds true for almost all even n.D.H.Lehmer (see [114]) showed that the aliquot sequence {s(k)(138)} after reaching a maximum s(117)(138) = 2 · 61 · 929 · 1587569, terminated at s(177)(138) = 1.(14) Guy et al. (see [114]) used a program to discover that

~~s(746)(840) = 601 (15) and establishing a new record s(287)(840) for the maximum of a terminating sequence. Recently, M. Dickerman found a new record showing that s(583)(1248) reaches a~~

73 CHAPTER 1 maximum of a number of 58 digits, the period being 1075 for this number n = 1248. (16) For recent advances in aliquot sequences (of computational nature), see M. Benito and J. L. Varona [20]. In 1977 H. W. Lenstra [188] has proved that it is possible to construct arbitrarily long monotonic increasing aliquot sequences, i.e. for each k there is an n with

~~n < s(n)~~

P. Erdos¨ [91] proved that for all ﬁxed k and δ>0 and for all n,except a sequence of density zero one has s(n) i s(n) i (1 − δ)n < s(i)(n)<(1 + δ)n (18) n n for i = 1, k. s(i+1)(n) s(n) The set of n with > − ε for i = 1, k (for each ε>0 and k)has s(i)(n) n asymptotic density 1. (19) Erdos,¨ Granville, Pomerance and Spiro [93] proved a similar result: for each ε> 0, the set of n with (2)( ) ( ) s n < s n + ε s(n) n has asymptotic density 1. (20) Let S(k)(x) denote the number of odd integers m ≤ x, not in the range of the function s(k). Then there is a δ>0 such that

S(k)(x) x1−δ (21) uniformly for all positive integers k, and x > 0. The unitary aliquot sequences and unitary sociable numbers are deﬁned similarly, by considering s∗(n) = σ ∗(n) − n (22) where σ ∗(n) denotes the sum of unitary divisors of n. The analogue of (17) holds true here, too, proved by te Riele [249]: for each k there is an n such that

~~n < s∗(n)~~

~~74 PERFECT NUMBERS~~

te Riele pursued all unitary aliquot sequences for n < 105. The only one which did not terminate or become periodic was 89610. Later calculations [250] showed that this reached a maximum at s∗(568)(89610),and terminated at its 1129th term. (25) Unitary sociable numbers may occur rather more frequently than their ordinary counterparts. M. Lal, G. Tiller and T. Summers [185] found cycles of periods 3, 4, 5, 6, 14, 25, 39, 65. For example, (30,42,54) is a 3-cycle, while (1482,1878,1890,2142,2178) is a 5-cycle. Beck and Najar [13] considered the iterates of

∗ ∗ L−(n) = s (n) − 1, and (26) ∗ ∗ L+(n) = s (n) + 1.(27) They introduce ”reduced unitary sociable numbers” by considering the iterations ∗ ∗ of L−,and ”augmented unitary sociable numbers” by the same for L+.Weshall call these in a more natural way as unitary quasi-sociable, resp. unitary almost- sociable numbers,inaccordance with the used terminologies in the case of perfect and amicable numbers. Beck and Najar showed by a computer search that there are no unitary almost- sociable numbers n for n < 105.(28) The same is true for unitary quasi-sociable numbers less than

~~110000 (29)~~

Recently, the calculations were lifted up to 109,andone cycle has been obtained for both cases (see [219]). (30) Cohen [58] introduces in a similar manner infinitary sociable numbers.He finds eight infinitary aliquot cycles of order 4, three of order 6 and one of order 11.(31) This last is the only cycle of order less than 17 and least member less than a million. Y. Kohmoto (see [326]) has considered a generalization of the sociable numbers as follows. Let (a(n)) be a sequence defined by σ(a(n − 1)) a(n) = . m If this sequence becomes cyclic after k > 1 terms, it is then called an 1/m- m−1 n−1 sociable number of order k.Itiseasy to see that e.g. 2 Mn and 2 Mm are 1/2-sociable numbers of order 2, with Mn and Mm being distinct Mersenne primes. In [219] one finds certain remarks and numerical results for the bi-unitary, exponential cycles, as well as their quasi and almost variants. (32)

~~75 CHAPTER 1~~

Hagis [129] considered nonunitary aliquot sequences. A k-tuple of distinct natu- # # ral numbers (n0, n1,...,nk−1) with ni = σ (ni−1)(i = 1, k − 1) and n0 = σ (nk−1) is called a nonunitary k-cycle.(33) 6 Asearch was made for all nonunitary k-cycles with k > 2andn0 ≤ 10 .A 3-cycle was found: (619368, 627264, 1393551)(34) An open question is whether or not unbounded nonunitary aliquot sequences exist. For generalized aliquot sequences, see te Riele [253]. For the iteration of various arithmetic functions, see [255]. Finally we mention two open problems, one due to Alaoglu and Erdos¨ [3], the other one to Poulet [245]. Alaoglu and Erdos¨ propose the question of whether or not the sequences:

σ(n), σ (σ(n)), ϕ(σ(σ(n))), σ (ϕ(σ(σ(n)))), . . . (35) or ϕ(n), ϕ(ϕ(n)), σ (ϕ(ϕ(n))), . . . (36) converge to a periodic state, a fixed number, or tend to infinity for all n? Poulet conjectures that the sequence ( fk(n))k defined by f0(n) = n, f2k+1(n) = σ(f2k(n)), f2k(n) = ϕ( f2k−1(n)) is eventually periodic, for all n.(37)

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~~[313] A. M. Vaidya, Comment on: ”History of perfect numbers”,Ganita Bharati 1(3-4)(1979), 22.~~

~~[314] R. C. Vaughan, Adventures in arithmetic, or: how to make good use of a Fourier transform, Math. Intellig. 9(1987), no. 2, 53-60.~~

~~[315] B. Volkmann, A theorem on the set of perfect numbers,Bull. A.M.S. 62(1956), Abstract 180.~~

~~[316] R. W. Van der Waall, On a theorem of Leopoldt and on perfect numbers, Nieuw Arch. Wiskunde (3) 18(1970), 159-161.~~

~~[317] S. Wagon, Perfect numbers,Math. Intellig. 7(1985), no. 2, 66-68.~~

~~[318] Ch. Wall, Topics related to the sum of unitary divisors of an integer, Univ. of Tennessee, 1970.~~

~~[319] Ch. R. Wall, Anew unitary perfect number, Notices A.M.S. 16(1969), 825.~~

~~[320] Ch. R. Wall, Bi-unitary perfect numbers,Proc. A.M.S., 33(1972), 39-42.~~

~~[321] Ch. R. Wall, The ﬁfth unitary perfect number,Canad. Math. Bull. 18(1975), 115-122.~~

~~[322] Ch. R. Wall, On the largest odd component of unitary perfect numbers, Fib. Quart. 25(1987), 312-316.~~

~~[323] Ch. R. Wall, New unitary perfect numbers have at least nine odd components, Fib. Quart. 26(1988), 312-317.~~

~~[324] G. C. Webber, Nonexistence of odd perfect numbers of the form 2β α 2β1 2β2 2β3 3 p s1 s2 s3 , Duke Math. J. 18(1951), 741-749.~~

~~97 CHAPTER 1~~

~~[325] E. Weisstein, http://mathworld.wolfram.com/AmicablePair.html.~~

~~[326] E. Weisstein, http://mathworld.wolfram.com/SociableNumbers.html.~~

~~[327] E. Weisstein, http://mathworld.wolfram.com/UnitaryAmicablePair.html.~~

~~[328] E. Wirsing, Bemerkung zu der Arbeit uber¨ vollkommene Zahlen,Math. Ann. 137(1959), 316-318.~~

~~[329] B. F. Yanney, Another deﬁnition of amicable numbers and some of their relation to Dickson’s amicables, Amer. Math. Monthly, 30(1923), 311-315.~~

~~[330] A. and E. Zachariou, Perfect, semi-perfect and Ore numbers, Bull. Soc. Math. Grece` (N.S.) 13(1972), 12-22.~~

~~[331] Perfect numbers and imperfect mathematicians (Persian), Bull. Iranian Math. Soc. 9(1)(1981/82), 84-79.~~

~~[332] http://www.utm.edu/research/primes/mersenne/index.html.~~

~~98 Chapter 2~~

~~GENERALIZATIONS AND EXTENSIONS OF THE MOBIUS¨ FUNCTION~~

~~2.1 Introduction~~

In this chapter we will survey many generalizations in Number theory of the Mobius¨ function, as well as analogues functions which arose by the extensions of certain divisibility notions or product notions of arithmetical functions. Our study will include also Mobius¨ functions in Group theory, Lattice theory, Partially ordered sets or Arithmetical semigroups. Sometimes references to applications will be pointed out, too. This is a reﬁned and extended version of [164]. The classical Mobius¨ function is deﬁned by   1, if n = 1 µ( ) = (− )k, = · ...· , n  1 if n p1 p2 pk pi distinct primes (1) 0, if n is divisible by the squares of a prime

~~The function occurred implicitly in L. Euler ([67] paragraph 269) with the considerations of the ”Riemann zeta function”~~

~~∞ − 1 1 1 ζ(s) = = − , s 1 s n=1 n p p~~

~~99 CHAPTER 2 and the reciprocal formula~~

~~ ∞ 1 1 µ(n) = 1 − = (2) ζ( ) s s s p p n=1 n~~

However, its arithmetical importance was ﬁrst recognized by A. F. Mobius¨ [143] in 1832, with the discovery of a number of inversion formulae. Mobius¨ raised the following question: Given an arbitrary function f (z) and g(z) by

~~∞ n g(z) = an f (z ), (3) n=1 express f (z) in terms of the functions g(zn),say~~

~~∞ n f (z) = bn g(z ). (4) n=1~~

2 If f (z) = c1z + c2z + ... (i.e., a power series), this question may be treated as aproblem in formal power series, and it is easily seen that (bn)n≥1 are obtained from the given coefﬁcients (an)n≥1 by 1, if n = 1 ad b n = . (5) d 0, if n > 1 d|n

~~Equivalently, (5) can be written with the use of formal Dirichlet series as~~

∞ ∞ an bn = , s s 1 n=1 n n=1 n i.e. (an)n≥1 and (bn)n≥1 form Dirichlet inverse sequences. In particular, if an = 1for all n, then bn = µ(n) by (1), and we have (formally) ∞ g(z) = f (zn), n=1 equivalent to

~~∞ f (z) = µ(n)g(zn). n=1~~

~~100 THE MOBIUS¨ FUNCTION~~

By setting f (ez) = F(z) and g(ez) = G(z), equations (3) and (4) take a more convenient form ∞ G(z) = an F(nz), (6) n=1 ∞ F(z) = bn G(nz) (7) n=1 respectively. The particular case an ≡ 1, bn ≡ µ(n) has a curious history, and a number of rediscoverers (e.g. by P. Tchebyschef in 1851 [203]; P. Bachman in 1894 [5] were unaware of Mobius’¨ paper). The notation µ(n) was ﬁrst introduced in 1874 by F. Mertens. All these considerations were formal and questions of convergence were ne- glected. Surprisingly, the ﬁrst rigorous treatment of these matters appeared only in 1936 and 1937 in works by E. Hille and O. Szasz´ [108] and E. Hille [107]. They proved the following result: If an and bn are related by (5), and ∞ g(n) = am f (mn)(n = 1, 2, 3,...), (8) m=1 where the double series akbm f (kmn) is absolutely convergent, then k,m ∞ f (n) = bm g(mn) (9) m=1 This theorem leads to a symmetrical version of (6)-(7): The following two assertions are equivalent: ∞ (i) g(n) = f (mn)(n = 1, 2,...) (10) m=1 ∞ and nε| f (n)| < ∞ for some ε>0; n=1 ∞ (ii) f (n) = µ(m)g(mn)(n = 1, 2,...) (11) m=1 ∞ and nε|g(n)| < ∞ for some ε>0. n=1

~~101 CHAPTER 2~~

There is a need for caution here for the truth e.g. of (6)-(7) for general f and g 1 (an = 1, bn = µ(n)). Put g(n) = ; then f (n) = µ(m)g(mn) is identically n m zero, so that g(n) = f (mn).Inthe other direction, if f (n) = µ(n)/n, then m g(n) = f (mn) is identically zero and again (6)-(7) fails. m The following remark by Andrew Lenard [2] has importance in Beurling’s approach to the Riemann hypothesis (see H. Bercovici and C. Foias¸ [12]. Let ∞ l(x) = (−1)n−1 L(nx), n=1 where ∞ L(x) = µ(n)K (nx), n=1 equivalent (by (6)-(7)) to ∞ K (x) = L(nx)(x > 0) n=1 Since l(x) = K (x) − 2K (2x),weget ∞ K (x) = l(x) + 2K (2x) = l(x) + 2l(2x) + 4K (4x) = ···= 2k f (2k x). k=0 So, by the Mobius¨ inversion formula ∞ l(x) = (−1)n−1 L(nx) (12) n=1 is equivalent to ∞ ∞ L(x) = 2kµ(n)l(2knx) (13) k=0 n=1 An improvement of corollary (10)-(11) has been obtained by J. H. Loxton and J. W. Sanders [138]: Suppose that the coefﬁcients an are completely multiplicative (i.e.amn = am · an for all m, n,anda1 = 1), and that the series am f (mn) and am g(mn) are m m absolutely convergent.

~~102 THE MOBIUS¨ FUNCTION~~

Then ∞ g(n) = am f (mn) (14) m=1 is equivalent to ∞ f (n) = µ(m)am g(mn) (15) m=1 Another application of the equivalence of (6)-(7) is related to the inversion of Fourier transforms. The following result is due to R. R. Goldberg and R. S. Varga [78]. Let k :[0, R] → R (R > 0) be of bounded variation, and suppose that

∞ |k(u)| log udu < ∞ 1 and put ∞ K (t) = k(u) cos tudu. 0 Then ∞ 1 K (0) nπ H(t) = + (−1)n K t 2 n=1 t is finite almost everywhere (0 < t < ∞), and ∞ k(t) = µ2n−1 H((2n − 1)t) (16) n=1 almost everywhere for t ∈ (0, ∞). In [78] it is noted that a similar result was obtained by R. J. Duffin in 1941. The ”finite” form of the Mobius¨ inversion formula n g(n) = f (d) ⇐⇒ f (n) = µ(d)g( ) (17) d|n d|n d wasgiven simultaneously by R. Dedekind [60] and J. Liouville [136] in 1857. Certain Mobius¨ function identities are important also in Sieve theory (see H. Hal- berstam and H.-E. Richert [84]). Identity (17) is a characteristic property of the Mobius¨ function, see S. Swetha- ranyam [200] and U. V. Satyanarayana [169].

~~103 CHAPTER 2~~

Interesting generalized Mobius¨ inversion formulae have been obtained in 1963 by D. E. Daykin [58]. ( ) = ( )( = ) ⇔ ( ) = Relation (17) can be written also as g n f x dx n f n µ(e)g(y)(ey = n) for n ≥ 1. Let D, E, X, Y be non-empty sets of positive integers and let k : N → R be an arithmetic function. Given a pair ( f, g) of arithmetic functions, we deﬁne another pair (F, G) by the relations G(n) = f (x)(dx = n; d ∈ D, x ∈ X), (18) F(n) = k(e)g(y)(ey = n; e ∈ E, y ∈ Y ) (19)

~~Empty sums take the value 0. We say that (D, E, X, Y, k) form a Mobius¨ system if, for all pairs ( f, g) of arithmetic functions satisfying~~

~~f (n) = 0 (n ∈ X) and g(n) = 0 (n ∈ Y ) (20) one has~~

~~F(n) = f (n) ⇔ G(n) = g(n) (21)~~

If A, B are sets of integers, let A· B denote the set of all products a ·b with a ∈ A, b ∈ B (i.e. Minkowski product). The following result is true ([58]): (D, E, X, Y, k) is a Mobius¨ system iff both H(n) = δ(n), where H(n) = k(e)(de = n; d ∈ D, e ∈ E), (22) and

~~X = Y = X · D = X · E. (23)~~

AsetC of integers is called a product set if C can be generated multiplicatively from a sequence (which could be ﬁnite, inﬁnite, or empty) (pi ) of pairly coprime positive integers, such that µ(pi ) =−1 for all i.Weassume that empty products take the value 1, and that 1 is in every product set. Moreover, the trivial set {1} is a product set. The following generalization of the Mobius¨ inversion formula holds true: Suppose that

~~n ∈ D ⇔ n ∈ E whenever k(n) = 0(24)~~

104 THE MOBIUS¨ FUNCTION and that k(n) = 0 whenever n has a square factor. Then (22) holds iff D is a product set, D = E and k(n) = µ(n) for n ∈ D. Let D ={1, 2, 8}, E ={2α : α ≥ 0}, X = Y = N, k(1) = k(4) = 1, k(2) =−1, k(2α)+k(2α−1)+k(2α−3) = 0 (α ≥ 3). Then (D, E, X, Y, k) isaMobius¨ system. If D is an isomorphic image of N ={1, 2,...} (which is a multiplicative semigroup) and k is the function induced by µ through the isomorphism, then (D, D, N, N, k) is a Mobius¨ system. The particular case of the isomorphism i → i k (k fixed positive integer) has been considered by P. J. McCarthy [34]. Forageneralization of an inversion formula due to Vinogradov, see R. T. Hansen [89]. A unified treatment of the Mobius¨ inversion formula appears in H. Breitenfellner [21]. A difference-operatorial approach to the inversion formulae is given in L. C. Hsu [114]. We note that the Mobius¨ function of a set of primes P was considered by A. k Wintner in 1943 [220]. This is a multiplicative function µP such that µP (p ) =−1 for p ∈ P and k = 1 and 0, otherwise. Wintner proved that if 1/p is divergent, p∈P ∞ then µP (n)/n = 0. n=1 The formula (17) exemplifies the combinatorial nature of the Mobius¨ function. In 1964 G. C. Rota [161] and his cowerkers have developed a general theory of Mobius¨ function on partially ordered sets, which includes the principles of inclusion- exclusion and Mobius¨ inversion as special cases (but, as we shall see in the following sections, many ideas and results were anticipated by S. Delsarte [61] or R. Wiegandt [219]). Formany results on various estimates, inequalities, as well as many related notions and functions (as the Mertens function M(x) = µ(n))onthe classical n≤x Mobius¨ function, see our Handbook I ([142]). This Chapter has, among many other things, also the aim to call the attention to many rediscoveries in the subject. But the main goal will be to survey the most important notions and results which involved the implications of Mobius¨ type functions (1). We will survey many generalizations in Number theory of the Mobius¨ function, as well as analogues functions which arose by the extensions of certain divisibility notions or product notions of arithmetical functions. Our survey will include also Mobius¨ functions in Group theory, Lattice theory, Partially ordered sets or Arithmetical semigroups. Sometimes references to applications will be pointed out, too.

~~105 CHAPTER 2~~

~~2.2 Mobius¨ functions generated by arithmetical products (or convolutions)~~

1Mobius¨ functions defined by Dirichlet products The Dirichlet product ”*” of two arithmetic functions f and g is defined by n ( f ∗ g)(n) = f (d)g( ) (1) d|n d where d | n denotes that d runs through all divisors d of n.The operation ”*” was considered as a binary operation on the set F of all arithmetical functions only at the beginning of the 20th century. M. Cipolla and E. T. Bell [10] proved first that (F, +, ∗) is an integrity domain. E. D. Cashwell and C. J. Everett [38] proved that this is in fact a factorial ring. Let F1 ={f ∈ F | f (1) = 0}, MF ={f ∈ F | f multiplicative and f ≡ 0}.Itcan be proved that (F1, ∗) is a commutative group, while (MF, ∗) is a subgroup of this Abelian group. In what follows, we shall use more times the arithmetical functions defined by

~~k Ek(n) = n , E(n) = n, e(n) = 1 (n ∈ N) (2) and 1, n = 1 δ(n) = . (3) 0, n > 1~~

Clearly e(n) = E0(n). By using the above group terminolgy, the inverse of the function e is exactly the Mobius¨ function µ: (as δ is the unity element of this group: f ∗ δ = δ ∗ f = f ), e ∗ µ = µ ∗ e = δ. We note that for the classical arithmetical functions ϕ,σ,d (representing Euler’s totient, the sum of the divisors and the number of the divisors, respectively) we have

~~ϕ = µ ∗ E,σ= E ∗ e, d = e ∗ e (4) or more generally~~

Jk = µ ∗ Ek,σk = Ek ∗ e, dk = dk−1 ∗ e (k ≥ 2) (5) where Jk is Jordan’s totient; σk is the sum of k-th powers of divisors; and dk is the Piltz divisor function (for more details, see e.g. [3], [90], [36], [177], [162]).

~~106 THE MOBIUS¨ FUNCTION~~

~~In 1963 in a little known paper, C. Popovici [151] introduced and studied a generalized Mobius¨ function, deﬁned by~~

µ = µ ∗ µ ∗ ...∗ µ k (6) k where k ≥ 1. It is immediate, by induction that k µ (pr ) = (−1)r (7) k r k where the binomial coefficient = 0 for k < r. Popovici proved more properties, r e.g. that µk is multiplicative; µk−1 = µk ∗ e, etc. Certain combinatorial applications, group theoretical considerations, as well as extensions of (6) to other functions in place of µ, can be found in a paper by E. Schwab and E. D. Schwab [174]. In 1998 L. C. Hsu and J. Wang [115] rediscovered this function. By a generalized α binomial coefficient for complex number α, via (7) this Mobius¨ function can be r extended, by α r r µα(p ) = (−1) (8) r and the assumption that µα is multiplicative, i.e. α ν (n) µα(n) = (− ) p 1 ν ( ) (9) p|n p n where νp(n) denotes the power of the prime number p in the factorization of n.This definition, as well as the repetition of some known properties are included in a paper by T. C. Brown, L. C. Hsu, J. Wang and P. J-S. Shiue [23]. We note that in 1996 R. G. Buschman [25] has introduced also µk for a positive integer k,byanother definition, namely as the Dirichlet inverse of the Piltz divisor function dk (in fact dk = µ−k). He has remarked that Mobius¨ generalized functions in this way were introduced also by E. D. Cashwell and C. J. Everett [38], and even by R Vaidyanathaswamy [210]. See also H. Scheid [170] and P. J. McCarthy [36]. More recent investigations by us reveal that µk wasinfact first introduced in the literature in 1915 by A. Fleck [69]. For applications of µk in the theory of Diophan- tine Equations, see R. Tschiersch [208].

~~107 CHAPTER 2~~

~~Extensions of µk by regular convolutions will be studied later. In 1968 D. Rearick [155] deﬁned the k-th power of an arithmetic function by~~

~~k = ∗ ∗ ...∗ . f f f f (10) k~~

Clearly, for f = µ,by(6) one reobtains µk. Let A denote the set of all real valued arithmetic functions, and let P denote the subset of A consisting of arithmetical functions f with f (1)>0. Rearick [155] deﬁnes the logarithm operator Log : P −→ A by

~~Log f (1) = log f (1),~~

1 n f (n) = f (d) f −1 d,(n > ) Log ( ) log 1 (11) f n d|n d where f −1 is the Dirichlet inverse of f . It can be proved that Log is a bijection of P onto A; therefore it is possible to deﬁne the exponential operator

~~Exp : A −→ P, Exp = Log−1. (12)~~

~~Let f ∈ P. The real power of f , i.e. f α (α ∈ R)isdeﬁned by~~

~~f α = Exp(αLog f ). (13)~~

~~Then, it can easily be shown that~~

−k = −1 ∗ −1 ∗ ...∗ −1 f f f f k where k is a positive integer. T. Caroll and A. A. Gioia [33] determined explicitely the powers f α of com- 1 1 pletely multiplicative functions for α =− and α = , and for positive rational 2 2 numbers α.In1997 P. Haukkanen [97] proved the following surprising fact: If f is a completely multiplicative function, then

~~α f = µ−α · f (14) for all real numbers α,where µ−α is the generalized Mobius¨ function of (8), (9) (put ”−α”inplace of ”α”).~~

~~108 THE MOBIUS¨ FUNCTION~~

Forarecent application of the function µα,see V. Laohakosol, N. Pabhapote and N. Wechwiriyakul [130]. By applying (14), in [130] the following is proved: Let f be a nonzero multiplicative function and α a nonzero real number. Then f is completely multiplicative iff

~~−1 (µα f ) = µ−α f (15)~~

~~For α = 1 this gives a result of T. M. Apostol (see [2], p. 49). Let us assume now that if α is a negative even integer, then~~

~~f (p−α−1) = f (p)−α−1 for each prime p. (16)~~

~~Let f be a nonzero multiplicative function and α ∈ R \{0, 1}.~~

~~α Assuming condition (16), if f = µ−α f, then f is completely multiplicative. (17)~~

For details, see [130]. Finally note that the Dirichlet product of arithmetic functions can be extended to functions f (n1,... ,nk) of k variables: ( f ∗ g)(n1,... ,nk) = f (a1,... ,ak)g(b1,... ,bk) (18)

~~a j b j = n j j = 1,... ,k~~

~~The arithmetic function 1, n = ...n = 1 δ (n ,... ,n ) = 1 k k 1 k 0, otherwise is the identity element for this operation. In 1975 Paula A. Kemp [123] deﬁned a Mobius¨ function of n arguments,~~

µ∼ (n1,... ,nk) = µ(n1) ···µ(nk) (19) and proved an inversion theorem. The theory of such functions, however (including (19)) was ﬁrst extensively studied by R. Vaidyanathaswamy [210] (see sec. 4 for (19)). See also a paper by M. V. Subbarao [191] (pp. 261-264).

~~109 CHAPTER 2~~

~~2 Unitary Mobius¨ functions The unitary product of two arithmetical functions f, g ∈ F is deﬁned by~~

~~n ( f g)(n) = f (d)g (20) d|n d ( , n )= d d 1~~

n i.e. one consider a sum on divisors d of n with d, = 1. The idea of a such d convolution is due to R. Vaidyanathaswamy [210] again, but its extensive study was initiated by E. Cohen [47], [48]. It can be shown that (F, +, ) is a commutative ring with unity element, (F, ) is an Abelian group having as (MF, ) as a subgroup. The function δ is also the unity element, while the inverse of e will be denoted, by analogy, by µ∗,asbeing the Mobius¨ function. In fact µ∗(n) = (−1)ω(n), (21) where ω(n) denotes the number of distinct prime divisors of n.Byanalogy with (4), one has ϕ∗ = µ∗ e, σ ∗ = E e, d∗ = e e, (22) ∗ = ∗ ,(≥ ). dk dk−1 e k 2 In [173] A. Schinzel proved that if f (1) = 1, the inverse function of f under the unitary convolution exists and is given by the formulae g(1) = 1and

ω(n) k k g(n) = (−1) f (di ), k=1 d1d2···dk =n i=1 for n > 1, where ω(n) is the number of distinct prime factors of n. For many properties of these functions, we quote only E. Cohen [47], [48], M. V. Subbarao [191], D. Suryanarayana [194] [195], J. Chidambaraswamy [43], J. Sandor´ [162], J. Sandor´ and L. Toth´ [166] (see also the monograph Mitrinovic-S´ andor-Crstici´ [142]). In 1971 E. M. Horadam [110] has extended Daykin’s Mobius¨ inversion formula for unitary divisors. Let D, E be non-empty sets of positive integers. Let k be an arithmetic function such that k(n) =±1 whenever n ∈ E. (23)

~~110 THE MOBIUS¨ FUNCTION~~

Consider the statement H(n) = δ(n), where H(n) = k(e)(de = n,(d, e) = 1, d ∈ D, e ∈ E). (24) AsetC of positive integers is called a unitary product set if it can be generated multiplicatively from a finite or infinite set of positive integers (pi ), mutually coprime. A general member of the set of generators will be denoted by pa(i).Ifn ∈ C, α α = 1 ... k then n has the form n pa(1) pa(k),but as we shall mainly be concerned with the evaluation of H(n),weshall take n = pa(1) ...pa(k). µ∗ ( ) = µ∗ ( ) = The unitary Mobius¨ function of the set C is defined by C 1 1, C pa(i) −1, and if n = pa(1) ...pa(k), then µ∗ ( ) = (− )k. C n 1 (25) Now, related to the statement (24), the following is true: Suppose that (23) holds true. Then (24) is true if and only if D is a unitary product set, D = E,and ( ) = µ∗ ( ) ≥ . k n D n for all n 1 (26)

3 Bi-unitary Mobius¨ function We introduce the bi-unitary convolution formula (see [100]). A divisor d > 0of the positive integer n is called bi-unitary if the greatest common unitary divisor of n d and is 1. We denote by (a, b) the greatest unitary divisor of both a and b. The d 1 bi-unitary product of two arithmetical function f, g ∈ F is deﬁned by

n ( f g)(n) = f (d)g . (27) d|n d ( , n ) = d d 1 1 The properties of these convolutions appear in [100]. If we consider the greatest common bi-unitary divisor of a and b and denote by n (a, b) ,wecalld triunitary divisor of n if (d, ) = 1. A. Bege introduced in [8] the 2 d 2 following generalization of Mobius’¨ function:  , =  1 if n 1 µ∗∗( ) = ∞ α n pα− p (28)  2k (−1)pαn k=1 , if n > 1 and proved the following inversion formula. If f, g ∈ F we have

~~n g(n) = f (d) ⇐⇒ f (n) = µ∗∗(d)g . (29) d|n d|n d ( , n ) = ( , n ) = d d 1 1 d d 1 1~~

~~111 CHAPTER 2~~

~~4Mobius¨ functions generated by regular convolutions~~

The regular convolution will be a common generalization of the Dirichlet and the unitary product. Let A(n) be a subset of the set of positive divisors of n. The A-convolution of two arithmetic functions f and g is deﬁned by

~~n ( f ∗A g)(n) = f (d)g . (30) d∈A(n) d~~

In 1963 W. Narkiewicz [145] deﬁned an A-convolution to be regular if (a) the set of arithmetical functions forms a commutative ring with unity with respect to the ordinary addition and A-convolution; (b) the A-convolution of multiplicative functions is multiplicative; (c) the function e has an inverse µA with respect to the A-convolution, and µA(n) = 0or−1 whenever n is a prime power. It is known that ([145]) an A-convolution is regular iff: (i) The conditions d ∈ A(n), m ∈ A(n) and m n d ∈ A(n), ∈ A d d are equivalent; n (ii) d ∈ A(n) implies ∈ A(n); d (iii) {1, n}⊆A(n); (iv) A(mn) ={de | d ∈ A(m), e ∈ A(n)},whenever (m, n) = 1; a a (v) for each prime power p (a ∈ N) there exists a positive integer t = tA(p ) such that

A(pa) ={1, pt , p2t ,... ,pst}, where st = a and pt ∈ A(p2t ), p2t ∈ A(p3t ),... ,p(s−1)t ∈ A(pa). Forexample, the Dirichlet convolution D = ” ∗ ” will be obtained by D(n) = the set of all positive divisors of n, while the unitary convolution U = ” ” follows n from U(n) ={d > 0 | d | n, d, = 1}.Itisimmediate that these convolutions d are regular. In what follows we shall assume that A is a regular convolution. The generalized Mobius¨ function µA is the multiplicative function (introduced by W. Narkiewiecz

~~112 THE MOBIUS¨ FUNCTION~~

~~[145]) given by   1, if a = 0 µ ( a) = − , a > . A p  1 if p 1isA-primitive (31) 0, if pa is non A-primitive~~

~~We note that a positive integer n is said to be A-primitive, if A(n) ={1, n}. Let~~

~~k k Ak(n) ={d ∈ N | d ∈ A(n )}.~~

In 1978 V. Sita Ramaiah [178] considered Ak-convolutions. The corresponding µ Mobius¨ function Ak therefore may be named as ”Sita Ramaiah’s Mobius¨ function”. He proved that the Ak-convolution is regular whenever the A-convolution is regular. In 1995 L. Toth´ [205] deﬁned the notion of a ”cross-convolution”. Let A be a regular convolution. Then A is named as a cross convolution if for every prime p we have either

~~a a 2 a a tA(p ) = 1 ( i.e. A(p ) ={1, p, p ,... ,p }≡D(p ), for every a ∈ N), or~~

~~a a a a tA(p ) = a ( i.e. A(p ) ={1, p }≡U(p ), for every a ∈ N).~~

It follows by the work of Sita Ramaiah [178] that if A is a cross convolution, then Ak = A for every k.For arithmetical functions involving cross convolutions, see L. Toth´ [206] and L. Toth´ andJ.Sandor´ [207]. Let now S be an arbitrary subset of N and A aregular convolution. Let S be the characteristic function of the set S, i.e. 1, if n ∈ S (n) = . S 0, if n ∈ S

~~P. Haukkanen [93] deﬁned the Mobius¨ function µS,A by~~

~~µS,A ∗A e = S, (32) i.e. µS,A(d) = S(n). d∈A(n)~~

If S ={1}, then µS,A ≡ µA-the Narkiewiecz Mobius¨ function. We note that the function µS,D (where D is the Dirichlet convolution) was introduced by E. Cohen in 1959 [46], and considered also by M. V. Subbarao and V. C. Harris [192].

~~113 CHAPTER 2~~

Other interesting particular cases of the Haukkanen Mobius¨ function appear in [93]. For example, let us deﬁne n to be a square with respect to the A-convolution if a a a a+1 a for each p n (i.e. p = 1, p | n, p n), we have a = 2st, t = tA(p ).Now, if S is the set of all squares with respect to the A-convolution, then µS,A = λA, where 1, if n = 1 λ (n) = +...+ a t , A (− )a1 ar , = 1 1 ··· ar tr 1 if n p1 pr

= ( ai ti ) λ with ti tA pi .Wenote that A is a generalized Liouville function. In [93] a generalized Ramanujan sum is considered, too. Following the paper [98] we say that an integer n > 1is(A, r)-powerful if for each A-primitive prime power pt ∈ A(n) we have θ(pt ) ≥ r and prt ∈ A(n), where

~~t st θ(p ) = sup{s ∈ N | tA(p ) = t}.~~

Let A be a regular convolution. An arithmetical function f is said to be A- n multiplicative (see K. L. Yocom [221]) if f ≡ 0and f (n) = f (d) f whenever d d ∈ A(n). An arithmetical function f is said to be A-rational function of order (s, r) if there exist A-multiplicative functions f1,... , fs and g1,... ,gr such that

~~−1 −1 −1 f = f1 ∗A f2 ∗A ...∗A fs ∗A (g1) ∗A (g2) ∗A ...∗A (gr ) (33)~~

This notion in case of A = D was introduced by R. Vaidyanathaswamy [210]. A combinatorial interpretation of a generalized Euler function, involving A- rational arithmetic function, and the A-extensions of the Fleck-Popoviciu-Buschman- Hsu-Wang Mobius¨ function is contained in [99]: µ = (µ ) ∗ (µ ) ∗ ...∗ (µ ) . A,k A A A A A A (34) k

~~Another Mobius¨ function µB, which is an analogue of µA appears in [99].~~

~~5 K -convolutions and Mobius¨ functions. B convolution The K -convolution or Davison-convolution of the arithmetical functions f and g is deﬁned by~~

n ( f ◦ g)(n) = f (d)g K (n, d), (35) d|n d where K is a complex valued function on the set of all ordered pairs (n, d) such that n is a positive integer and d is a positive divisor of n. The concept of K -convolution

114 THE MOBIUS¨ FUNCTION originates to T. M. K. Davison [56] from 1966. In the case in which K (n, d) depends n only on the g.c.d. (d, ), the concept is due to A. A. Gioia [75] and A. A. Gioia and d M. V. Subbarao [76]. Further study of K-convolutions is included in M. Ferrero [68], I. P. Fotino [72], M. D. Gessley [74], P. J. McCarthy [35], P. Haukkanen [92]. It is known (see the above References) that the set MF of multiplicative arithmetical functions forms an Abelian group with identity with respect to the K convolution iff (a) K (n, n) = K (n, 1) = 1 for all n; (b) K (mn, de) = K (m, d)K (n, e) for all m, n, d, e such that d | m, e | n,(m, n) = 1; (c) n d K (n, d)K (d, e) = K (n, e)K , e e for all n, d, e such that d | n, e | d; (d) n K (n, d) = K n, d for all n, d with d | n. Forexample, the regular convolution by Narkiewcz satisﬁes (a)-(d). The k-th K -iterate of an arithmetical function f is deﬁned by

(k) = ◦ ◦ ...◦ f f f f (36) k Clearly, this is a generalization of Mobius’¨ function considered in former paragraphs (see e. g. (10), (35)). Clearly (k) f (n) = f (a1) ··· f (ak)K (n, a1)K (a2 ···ak, a2) ···K (ak−1ak, ak−1) a1a2···ak =n (37)

~~The inverse of an arithmetic function f with respect to the K -convolution is deﬁned by~~

~~f ◦ f (−1) = f (−1) = δ.~~

~~The inverse exists and is unique iff f (1) = 0 (see [56]).~~

~~115 CHAPTER 2~~

In paper [94], among other results, the following is proved: Suppose f is an arithmetical function such that f (1) = 1. Then f (r) is multiplicative iff f is multiplicative. Questions of ”quasi-multiplicativity” as well as ”semi-multiplicativity” are also considered (see also D. Rearick [154]). An interesting particular case, introduced in the book by P. J. McCarthy ([36], p. ν ( ) = p n 168) is the so called ”binomial convolution”. For n p|n p deﬁne the function B(n, d) as ν (n) B(n, d) = p ν ( ) (38) p|n p d a where ν (n) denotes the power of the prime p in the canonical form of n, and p b is a binomial coefﬁcient. The binomial convolution follows from (35), when K ≡ B. In what follows, we shall use the notation ” ◦ ” = ” ◦B ” and the binomial convolution will be called simply as the ”B-convolution”. We note, that the B-convolution shares many basic properties with the Dirichlet convolution. For example, the set of all arithmetic functions f with f (1) = 0 forms a group under the B-convolution, and the set of all multiplicative functions forms a subgroup of this group. On the other hand, the B-convolution has properties which differ from those of the Dirichlet convolution. A fundamental diference is that the B- convolution preserves complete multiplicativity, while the Dirichlet product of two completely multiplicative functions generally is not completely multiplicative (but it is ”specially multiplicative” (see [36], p. 24 and [181])). The following interesting theorem is proved in [96]. We have f (n) = g(d)B(n, d) ⇐⇒ g(n) = f (d)λ(d)B(n, d). (39) d|n d|n where

λ(n) = (−1) (n) is the Liouville function. This means that the inverse e−1 of the function e,related to a B-convolution, is the Liouville function. In other words, µ ( ) = λ( ) ◦B n n (40) i.e. the Liouville function appears here as the Mobius¨ function generated by the B-convolution.

~~116 THE MOBIUS¨ FUNCTION~~

The B-convolution has also an interesting connection to the convolution of sequences. It is well known that the binomial convolution of sequences (an)n≥0, (bn)n≥0 is a sequence (cn)n≥0 given as n n cn = akbn−k (41) k=0 k (see R. L. Graham, D. E. Knuth and O. Patashnik [80], section 7.6). If f and g are multiplicative functions, then their binomial convolution f ◦B g is completely determined by their values at all prime powers pn.For a ﬁxed prime p, n n let f (p ) = an and g(p ) = bn for all n ≥ 0. Then n n n k n−k ( f ◦B g)(p ) = f (p )g(p ) = = k k 0 n n = akbn−k = cn k=0 k This is similar to the connection between Dirichlet convolution and Cauchy product

~~n akbn−k. k=0 Forrelated questiones, involving e.g. ”roots of sequences” under the Dirichlet and binomial convolution, see [95].~~

6 Exponential Mobius¨ functions The exponential convolution was deﬁned by M. V. Subbarao in 1972 [191] by   ( f ∗ g)(1) = f (1)g(1)  ex  n1 nk (42)  ( ∗ )( ) = ... d1 ··· dk d1 ··· dk  f ex g n f p1 pk g p1 pk d1|n1 d2|n2 dk |nk

= n1 n2 ··· nk where n p1 p2 pk is the prime factorization of n. It is immediate that this convolution is commutative and associative. This is not n of Narkiewicz type, not being of the form f (d)g . d|n d Subbarao proved the following results: 1) (F, ∗ex) is a commutative semigroup with identity element |µ| (where µ is the classical Mobius¨ function);

~~117 CHAPTER 2~~

2) The units of (F, ∗ex) are those functions f for which f (n) = 0 whenever n is aproduct of distinct primes and f (1) = 0; 3) The semigroup (F, ∗ex) has an inﬁnity of zero divisors. An element f of F, ∗ex) is a non zero divisor only if, given any ﬁnite number of primes p1,... ,pr , there exist coresponding positive integers a1,... ,ar such that a1 · ...· ar = ; f p1 pr 0

~~4) (F, ∗ex) has no nontrivial nilpotent elements. He also deﬁned the exponential analogue µ(e) of the Mobius¨ function as follows:~~

~~µ(e)(1) = 1; (e) µ (n) = µ(a1) · ...· µ(ar ), (n > 1) (43)~~

~~= a1 · ...· ar . where n p1 pr Then µ(e) is multiplicative and e-multiplicative (i.e. satisﬁes the functional equation~~

~~f (pab) = f (pa) f (pb), for (a, b) = 1and all primes p,where f is multiplicative). The exponential inverse of the function given by e(n) = 1,(n ∈ N) will be exactly µ(e). Therefore~~

~~(e) f = g ∗ex e ⇐⇒ g = f ∗ex µ . (44)~~

Let now A be a regular convolution of Narkiewicz type. The exponential A- convolution of the functions f and g has been introduced by J. Hanumanthachari [87] and K. Shindo [176]:   ( f ∗ g)(1) = f (1)g(1)  Aex  n1 nk  ( ∗ )( ) = ... d1 ··· dk d1 ··· dk  f Aex g n f p1 pk g p1 pk d1∈A(n1) d2∈A(n2) dk ∈A(nk ) (45)

= n1 n2 ··· nk where n p1 p2 pk is the prime factorization of n. Hanumanthachari proved that (F, ∗Aex ) is a commutative semigroup with identity |µ|. Units are those functions f , for which f (γ (n)) = 0, where γ(n) = p1 p2 ···pk is the greatest squarefree divisor (or the ”core”) of n.Further, it can be veriﬁed that the set of all exponentially multiplicative functions forms an Abelian

118 THE MOBIUS¨ FUNCTION group under the exponential A-convolution. For example, the inverse of the function −1 = µ(e) e is e A ,given by µ(e)( ) = ; A 1 1 µ(e)( ) = µ ( ) · ...· µ ( ), ( > ) A n A n1 A nk n 1 (46) = n1 · ...· nk . where n p1 pk µ(e) Clearly, A is multiplicative and exponentially multiplicative. For the exponential totient function, see J. Sandor´ [163], P. Haukkanen and P. Ruokonen [103] and for a survey of results, including properties on asymptotics, see Mitrinovic-S´ andor´ [142]. Exponentially A-multiplicative functions, as well as their Apostol type characterizations, appear in [103]. For example the following result is proved: An exponentially multiplicative function f is exponentially A-multiplicative (i.e. n d n f (γ (n)) = 0 and f (p ) = f (p ) f (p d ) for all primes p and all d, n with d ∈ A(n)) iff −1 = µ(e) , f A f (47) µ(e) where a is given by (46). Foranotion and results related to strong regular A-convolution,seeA.C. Vasu, V. V. Subrahmaya Sastri and C. S. Venkataraman [211].

7 l.c.m.-product (von Sterneck-Lehmer) The von Sterneck-Lehmer, or l.c.m.-product, appeared in R. D. von Sterneck [190] and studied by D. H. Lehmer [131], is deﬁned by ( f g)(n) = f (k)g(m) (48) [k,m]=n where [k, m] denotes the l.c.m. of k and m. von Sterneck proved the following connection between l.c.m. product and Dirich- let product: ( f g) ∗ e = ( f ∗ e) · (g ∗ e) (49) where e(n) = 1, ∀n ∈ N,and”·”isthe pointwise product. From this relation directly follows that the l.c.m.-product of multiplicative functions is multiplicative. It can be veriﬁed that there exist non-trivial divisors of zero for this product. Let f be any function such that f (1) = 0 and let g = µ.Infact, H. Scheid [170] pointed out that it is easy to show that if f = δ, and f f = f,

~~119 CHAPTER 2 then f is a zero divisor. If however, (g ∗ e)(n) is non zero for all n, then from (49) it is an easy proof to show that~~

~~f g = 0 ⇒ f = 0.~~

~~Consequently, we can prove that if g is a non-negative function which has a Dirichlet inverse, then we have the cancellation property:~~

~~f g = h g ⇒ f = h. (50)~~

In order to state an inversion formula for the l.c.m. product, Lehmer [131] introduced the arithmetical function   1, if n = 1 ( ) = r r Ms n {( + )s − s}, = ar (51)  ak 1 ak if n pk k=1 k=1 where s ∈ R. Ms is a common generalization of δ and e, since

~~M0 = δ, M1 = e.~~

~~In fact~~

~~k Mk = e e ... e = µ ∗ d (k ≥ 0), (52) k see R. G. Buschman [29]. Lehmer proved the following inversion theorem of Mobius¨ type:~~

~~f = e g ⇐⇒ g = M−1 f. (53)~~

~~This means that M−1 provides the ”Mobius¨ function” analogue in the case of l.c.m. products:~~

~~µ ≡ M−1. (54)~~

~~We note that M−1 can be written also as r 1 1 (−1)ω(n) M− (n) = − = 1 + r k= ak 1 ak 1 ak(ak + 1) k=1~~

~~120 THE MOBIUS¨ FUNCTION~~

~~A list of l.c.m. products appears in [29]. We quote only the following interesting relations:~~

~~f δ = f f µ = f (1)µ ϕ ... ϕ = Jk k ϕ Jk = Jk+1~~

~~Mk Mm = Mk+m~~

~~Jh Jk = Jh+k µ (d∗ · λ) = λ.~~

8 Golomb-Guerin convolution and Mobius¨ function The Golomb-Guerin convolution will be defined as follows. Let F denote the set of elements of positive integers which are not perfect powers > 1 (thus 1 ∈ ≥ = a1 ··· ar = g F). Any integer n 2, n p1 pr can be expressed as n m , where g = g.c.d.(a1,... ,ar ) and m ∈ F.In1973 S. W. Golomb [79] defined the ”root function” r(n) equal the number of distinct representations n = ab,with a, b positive integers, and noted that r(n) = d(g) for n = mg,(m ∈ F, g ∈ N), with d denoting the number of distinct divisors. Fortwo arithmetical functions f, g and n = mg,(n ∈ F, g ∈ N),E.E.Guerin [81] defined the G-convolution as   ( ∇ )( ) =  f g 1 1 . (55)  d g  ( f ∇g)(n) = f m g m d d|g

Guerin [81] has proved the following result: (1) (F, +, ∇) is a commutative ring with unity δ∇ (where δ∇ (n) = 1 for n = 1 or n ∈ F; δ∇ (n) = 0 otherwise). (2) fisaunit in (F, +, ∇) iff f (1) = 0 and f (m) = 0 for all m ∈ F. (3) An arithmetic function f is a nondivisor of zero iff f (1) = 0 and for each m ∈ F there exists a positive integer g such that f (mg) = 0. He also showed that the set of G-multiplicative functions (i.e satisfying f (1) = 1 and f (mab) = f (ma) f (mb) for m ∈ F, and (a, b) = 1) which are units in (F, +, ∇) form an Abelian group.

~~121 CHAPTER 2~~

~~Now, as usual, the inverse of e with respect to ∇ will be named a Mobius¨ function. Put~~

~~e∇µ∇ = δ∇ , (56) i.e. 1, if n = 1 µ∇ (n) = . (57) µ(g), if n = mg (m ∈ F, g ∈ N)~~

This is the Guerin analogue of the Mobius¨ function. Euler’s totient and the sum of divisors functions will be introduced as ϕ ( ) = (µ ∇ )( ) = ∇ n ∇ E n g g = µ(d)m d (n = m (m ∈ F, g ∈ N)), (58) d|g

~~σ∇ (n) = e∇ E.~~

~~It can be veriﬁed that ϕ∇ and σ∇ are not G-multiplicative functions.~~

9 max-product (Lehmer-Buschman) The Lehmer-Buschman or max product is deﬁned by ( f ♦g)(n) = f (k)g(m) (59) max{k,n}=n introduced by D. H. Lehmer [133]. Lehmer proved that the ♦ product is associative, commutative and has δ as the identity. There are nontrivial divisors of zero. A fundamental identity proved by Lehmer is

~~( f ♦g)#e = ( f #e) · (g#e), (60) where the product ”#” is deﬁned by~~

x ( f #g)(x) = f (k)g , (61) k≤x k which has a name ”Cauchy product”. There exists a useful identity which connects Dirichlet and # products (see T. M. Apostol [3], I. Niven-H. S. Zuckerman [148]):

~~f #(g#h) = ( f ∗ g)#h. (62)~~

~~122 THE MOBIUS¨ FUNCTION~~

As corollaries of (62) note that: (1) if f (1) = 0, then f #g = f #h ⇒ g = h; (2) if S is the greatest integer function f = e ∗ g ⇒ f #e = g#S; (3) f #(g#h) = g#( f #h). Left inversion does have the same form as the Dirichlet inversion, and it follows by the use of (62), but the order of factors is now important: f = e#g ⇐⇒ g = µ# f. (63) Right inversion is a different matter. We have f = g#e ⇐⇒ g = f, (64) where f (1) = f (1) and f (n) = f (n) − f (n − 1), (n ≥ 2). Since the inverse operator for the right #-multiplication by e is the difference operator ,wecan write another form for (60): (f )♦(g) = ( f · g). (65) By manipulating with the operator and equation (65), we can derive the right inversion formula of Mobius¨ type ([28]):

~~f = e♦g ⇒ g = (E−1)♦ f, (66) where 1 1 (E− )(n) = − ,(n > 1). 1 n n − 1 More generally, it can be proved [28]: = ♦ ...♦ ♦ ⇐⇒ = ( ) ♦ . f e e g g E−k f (67) k~~

~~Remark that letting in (66) f ≡ δ,wecan deduce a ”right-Mobius¨ function” µ♦ by~~

~~µ♦ = (E−1) ♦δ. (68)~~

~~123 CHAPTER 2~~

10 Infinitary convolution and Mobius¨ function The infinitary divisors have been introduced also in Chapter I (see paragraph 9). Aslightly distinct introduction can be done via the I -components of a number. Let α I ={p2 : p prime,α ≥ 0}.Itiseasy to see that n > 1 can be written in exactly one way (except the order of factors) as a product of distinct elements of I .Weshall call each element of I in this product an I -component of n, and we shall say that d is an I -divisor of n if every I -component of d is also an I -component of n.Wewrite d|∞n for an I -divisor d of n. Suppose that n = P1 P2 ...Pt , where P1 < P2 < ··· < Pt are the I -components of n. Put t = J(n) (let (J(1) = 0, by definition). The infinitary Mobius¨ function µ∞ is given by (see G. L. Cohen and P. Hagis, Jr. [53])

~~J(n) µ∞(n) = (−1) . (69)~~

~~It is immediate that µ∞(d) = δ(n). (70) d|∞n~~

~~The inﬁnitary convolution of two arithmetic functions f and g is deﬁned by~~

~~n ( f ∗ g)∞(n) = f (d)g . (71) d d|∞n~~

Then it follows that the set of arithmetic functions f with f (1) = 0 forms an abelian group with respect to the operation of inﬁnitary convolution, with identity element δ, i.e. δ(n) = 1, n = 1, δ(n) = 0, n > 1. The following Mobius¨ inversion formula holds true:

~~n F(n) = f (d) ⇔ f (n) = F µ∞(d). (72) d d|∞n d|∞n~~

~~Another inﬁnitary version of the Mobius¨ functions appears in sequence A064179 of N. J. A. Sloane’s Encyclopedia [183].~~

11 Mobius¨ function of generalized (Beurling) integers A. Beurling [13] defined the so-called ”generalized integers” as follows: Suppose there is given a finite or infinite sequence of real numbers (”generalized primes”) such that 1 < p1 < p2 < ....Fromthe set {l} of all possible p-products, i.e. products α α 1 ... r α ,...,α ≥ p1 pr with 1 r 0 integers of which all but a finite number are 0. Call

124 THE MOBIUS¨ FUNCTION these numbers generalized integers and suppose that no two generalized integers are equal if there α’s are different. Suppose also that {l} may be arranged as an increasing sequence

~~0 < 1 = l1 < l2 < l3 < ···< ln < ... (73)~~

~~Denote for simplicity ln = N, lr = R, ld = D etc. Then a notion of divisibility will follow by~~

~~D|N iff ∃ = lδ : N = D (74)~~

Let (R, S) denote the greatest common l-divisor of R = lr and S = ls. Similarly, (R, S)k denotes the greatest of the generalized integers, which are kth powers of generalized integers and which divide R and S. Therefore, if (R, S)k = 1, then R and S are k-prime to each other. E. M. Horadam [111], [112] deﬁned the Mobius¨ function of a generalized integer N as (−1)k, if N is squarefree among {l} µ(N) = (75) 0, otherwise V. V. Subrahmanyasastri [193] has introduced the generalization µ(N 1/k), if N is a kth power of a generalized integer, µ (N) = (76) k 0, otherwise

By deﬁning the convolution of two arithmetic functions deﬁned on generalized integers by N ( f · g)(N) = f (D)g (77) D|N d and using (75), as well as (76), Mobius-type¨ inversion formulae can be deduced. Similarly, various totient functions or Ramanujan sums can be introduced and studied (see e.g. [111], [112], [193]). Forasurvey of solved and unsolved problems on generalized integers see E. M. Horadam [113].

~~12 Lucas-Carlitz (l-c) product and Mobius¨ functions Let p be a prime and for the nonnegative integers n and r put~~

~~2 n = n0 + n1 p + n2 p + ... (0 ≤ n j < p),~~

~~2 r = r0 + r1 p + r2 p + ... (0 ≤ r j < p).~~

~~125 CHAPTER 2~~

Afamous theorem of E. Lucas [139] states that n n n n = 0 1 2 ...(modp) (78) r r0 r1 r2 n In particular, it follows from (78) that the binomial coefﬁcient = Cr is prime r n to p if and only if

0 ≤ r j ≤ n j ( j = 0, 1, 2,...) (79) We accordingly define the Lucas-Carlitz product [31] by n ∗ ( f ∗l−c g)(n) = f (r)g(n − r) (80) r=0 where the summation is restricted to those r that satisfy (79). Here f and g are arithmetic functions defined on the set of nonnegative integers. It is immediate from the definition that the l-c product is both associative and commutative; also it is distributive with respect to the usual addition of function. The function z(n) = 0 (n = 0, 1,...) is the identity for addition, while the function δ modified to δ0(n) = 1, n = 0; δ0(n) = 0ifn > 0 becomes the multiplicative identity for (80). If f ∗l−c g = δ0,wecall g the inverse of f . The following theorem holds (see [31]): An arithmetic function f possesses an inverse if and only if f (0) = 0. (81)

~~Particularly, the function e0 given by e0(n) = 1 (n = 0, 1, 2,...) satisﬁes the conditions, so has an inverse µl−c:~~

~~e0 ∗l−c µl−c = δ0 (82)~~

r If n = a0 + a1 p + ···+ar p (0 ≤ a j < p),itiseasily veriﬁed that   1ifa = 0 r µ ( r ) = − = µ ( ) = µ ( j ) l−c ap  1ifa 1 and l−c n l−c a j p (83) 0ifa > 1 j=0 where 0 ≤ a < p. The following inversion theorem is valid: n n ∗ ∗ g(n) = f (r) ⇔ f (n) = µl−c(r) f (n − r). (84) r=0 r=0

~~126 THE MOBIUS¨ FUNCTION~~

~~r If m, n are deﬁned by n = a0 + a1 p + ··· + ar p (0 ≤ a j < p) and m = r b0 + b1 p + ···+br p (0 ≤ b j < p) we deﬁne~~

~~r j (m, n)p = p min{a j , b j } (85) j=0 An arithmetic function f is called factorable if f (0) = 1and~~

~~f (m + n) = f (m) f (n) for (m, n)p = 0. (86)~~

~~Forexample, e0 and µl−c are factorable functions. Related to factorable functions the following general theorem holds true: Let h = f ∗l−c g. Then~~

~~if any two of the functions f, g, hare factorable the third is also. (87)~~

~~n ∗ As a corollary one deduces that if g(n) = f (r), then r=0~~

~~f is factorable ⇔ g is factorable. (88)~~

13 Matrix-generated convolution In what follows we shall consider matrix-generated convolutions and corresponding Mobius¨ functions. Suppose that G = (gij) is an inﬁnite dimensional matrix having elements 0 and 1: gij = 1ifi = j and gij = 0ifi > j, and that the 1’s in column n of G appear in rows n1, n2,...,nk (n1 < n2 < ···< nk = n). Let f, g be two arithmetic functions and deﬁne the convolution (see E. E. Guerin [82])

k ( f ∗G g)(n) = f (ni )g(nk+1−i ), n = 1, 2, 3,... (89) i=1 generated by the matrix G. Clearly, ∗G is a commutative operation on the set of arithmetic functions. Let Gn be the n × n submatrix of G = (gij) with 1 ≤ i ≤ n,1≤ j ≤ n. Examples. 1) The matrix D = (dij) with dij = 1ifi| j, dij = 0 otherwise, generates the Dirichlet convolution ∗D. Here Dn is the n × n divisor matrix, and the set {n1, n2,...,nk} is the set of positive divisors of n. 2) The unitary convolution is generated by the matrix U = (uij) with uij = 1if i ≤ j, and i| j and i and j|i are relatively prime; uij = 0 otherwise.

~~127 CHAPTER 2~~

~~3) The matrix C = (cij) deﬁned by cij = 1ifi ≤ j, cij = 0 otherwise, generates a convolution ∗C related to the Cauchy product. Since {n1, n2,...,nk}= {1, 2,...,n},wehave~~

( f ∗C g)(n) = f (1)g(n) + f (2)g(n − 1) +···+ f (n)g(1). = ( ) = ≤ 4) Fo rafixedprime p, let the matrix L lij be defined by lij 1ifi j j−1 = ∗ and p i−1 , lij 0 otherwise. The convolution L generated by L is related to the Lucas-Carlitz product. Define a general Mobius¨ function µG by

e ∗G µG = δ (90) where, as usual e(n) = 1 for all n and δ(n) = 1ifn = 1, = 0, n > 1. It is immediate −1 = × −1 = ( ) from Gn Gn In (the n n identity matrix) that if Gn gij , then = µ( ) gij j (91) for j = 1, 2,...,n (where µ is the classical Mobius¨ function). For example, the ele- −1 µ ( ) = µ( ), µ( ), ...,µ( ) ments in row one of D6 (see Example 1 above) are D 1 1 2 6 (in that order). Deﬁne now two arithmetic functions A and B by

~~n n ( ) = ( ), ( ) = ( ). A n gin f i B n ging i (92) i=1 i=1~~

Now suppose that G and ∗G satisfy the following two properties: (i) f ∗G δ = f for all arithmetic functions f ; (ii) ∗G is an associative operation on the set of arithmetic functions (E.g. in Ex- amples 1)-4) properties (i) and (ii) are valid). As from (92) one has A = e ∗G f and by (i), (ii)

~~( f ∗G g)(n) = ( f ∗G e ∗ B)(n), so~~

g(n) = (e ∗G B)(n) ⇔ B(n) = (g ∗ µG )(n) (93) which is a Mobius¨ inversion formula. For details, see [82]. We stop here the consideration of various convolution products. We note that other convolutions are included in the references (e.g. semi-unitary product [43], Lehmer ψ-product [132], [179], [147], partial product [26], integral product [27], convolutions with unbounded unity [180], etc.).

~~128 THE MOBIUS¨ FUNCTION~~

~~2.3 Mobius¨ function generalizations by other number theoretical considerations~~

1Apostol’s Mobius¨ function of order k Mobius¨ functions of order k, introduced by T. M. Apostol [2] in 1970 are deﬁned by the formulas   1, if n = 1,   0, if pk+1 | n for some prime p, α µk(n) = (− )r , = k ··· k i , ≤ α < , (1)  1 if n p1 pr pi with 0 i k  i>r 1, otherwise.

Among other results (e.g. µk is multiplicative) Apostol obtained an asymptotic formula like 1 µk(n) = Ak x + O(x k log x), (2) n≤x where 2 1 A = 1 − + . k k k+1 p p p By assuming the Riemann hypothesis, D. Suryanarayana [196] showed that, the error term in (2) can be improved to

4k O x 4k2+1 f (x) , (3) where f (x) = exp{A log x(log log x)−1} for some positive constant A.Afurther generalization of the Apostol function has been provided by A. Bege [9], who considered the application   1ifn = 1,   1ifpk n for each prime p, α µk,m(n) = (− )r = m ··· m i , ≤ α < , (4)  1 if n p1 pr pi with 0 i k  i>r 0 otherwise.

~~We remark that if m = k, µk,k(n) = µk(n). A. Bege [9] proved for x ≥ 3andm > k ≥ 2 the following asymptotic result:~~

~~2 xn αk,m 1 µ , (r) = + θ(n)x k δ(x) . k m ζ( )ψ ( )α ( ) 0 (5) r≤x k k n k,m n (r,n)=1~~

129 CHAPTER 2 uniformly in x, n and k, where 1 α , = 1 − ; k m m−k+1 + m−k+2 + ···+ m p p p p and 1 α , (n) = n 1 − ; k m m−k+1 + m−k+2 + ···+ m p|n p p p θ(n) = d∗(n) = the number of square-free divisors of n; 1 1 ψ (n) = n 1 + + ···+ ; k k−1 p|n p p where k is an integer ≥ 2,

3 − 1 δ(x) = exp {−A log 5 x (log log x) 5 }, with A > 0anabsolute constant. By assuming that the Riemann hypothesis is true, the following asymptotic formula holds 2 xn αk,m 2 µ , (r) = + θ(n)x 2k+1 f (x) . k m ζ( )ψ ( )α ( ) 0 (6) r≤x k k n k,m n (r,n)=1 uniformly in x, n and k, with f (x) = exp {A log x (log log x)−1},(A > 0). A main ingredient in the proof is the identity µk,m(n) = µ(d) qk(δ) (7) δdm =n (d,δ)=1 where k ≤ m, µ is the classical Mobius¨ function and qk is the characteristic function of the k-free integers.

2 Sastry’s Mobius¨ function Asomewhat similar Mobius¨ generalization has been considered by K. P. R. Sastry α = r i [167]. If n i=1 pi ,let   1, if n = 1, µk(n) = 0, if n contains a k-th power divisor (8)  α +α +...+α (−1) 1 2 r otherwise

~~130 THE MOBIUS¨ FUNCTION~~

~~Clearly, µ2(n) = µ(n). This function is multiplicative, and the sum Sk(n) = µk(d) d|n has the property:   0, if (i)α~~

As a corollary one obtains that Sk(n) vanishes iff one of the following holds: (i) n > 1 and admits a kth power divisor, and k is even, (ii) n > 1, k is odd and at lest one αi < k, and k is odd, (10) (iii) n > 1 and does not admit a kth power divisor, and one of αi is odd In every other case one has S (n) = 1. (11) k For the sum µk(n) for k = 1, 2,...,m one has m , µ ( ) = , 0 if m n 0 µk(n) = α (12) (−1) i (m − l + 1) if µ − (n) = 0,µ(n) = 0 and l ≤ m k=1 l 1 l The Dirichlet series of this function is ζ(2s) 1 ∞  · , for k even µ (n) ζ(s) ζ(ks) k = . (13) ns  ζ( ) ζ( ) n=1  2s · ks , ζ(s) ζ(2ks) for k odd Another property is related to the sum r β n i µ = (−1)i=1 , (14) k dk dk |n where αi ≡ βi (mod k) and 0 ≤ βi < k (1 ≤ i ≤ r). Particularly, if k is even,  r   1, if αi =even  n i=1 µ = . (15) k k  k | d  r d n   −1, if αi =odd. i=1

~~131 CHAPTER 2~~

3Mobius¨ functions of Hanumanthachari and Subrahmanyasastri In 1978 J. Hanumanthachari and V. V. Subrahmanyasastri [88] introduced a Mobius¨ function µs(M, N) as follows: Let s and M be any two positive integers; then for any positive integer n deﬁne 0, if N has a square factor µ ( , ) = , s M N (−(s + λ(p)), otherwise (16) p|N where, for any prime p, λ(p) = λ(M, p) is the number of k ∈{1, 2,... ,s} for which (M − k + j, p) = 1 for each j ∈{1, 2,... ,s}. In fact, this new Mobius¨ function arises in perceiving an extended Nagell totient function θs(M, N) in the light of the deﬁnition of a totient function of Vaidyanathaswamy [210]: The Dirichlet product of a totally multiplicative function and the inverse of another totally multiplicative function is called a totient. When s = 1, we get the function , µ ( , ) ≡ µ( , ) = 0 if N has a square factor , 1 M N M N ω( N ) ω(m) (−1) m · (−2) , otherwise (17) where ω(m) denotes the number of distinct prime factors of m; m being the greatest divisor of N relatively prime to M. When M ≡ 0 (mod N); (16) reduces to 0, if N has a square factor µ (N) = , (18) s (−s)ω(N), otherwise and this reduces to the classical Mobius¨ function when s = 1. Denote the Jordan-Nagell totient function by θ (s)(M, N), equal to the number of ordered s-tuples < x1,... ,xs > with x j (mod N), j ∈{1,... ,s} such that ((x j ), N) = 1 and ((M j − x j ), N) = 1, where (r j ) denotes the greatest common divisor of (r1,... ,rs). This is given by 1 2 θ s(M, N) = N s 1 − 1 − . (19) ps ps p|(M,N) p|N pM

~~132 THE MOBIUS¨ FUNCTION~~

The Schemmel-Nagell totient, denoted by θs(M, N) is equal to the number of s- tuples of consecutive integers < x + 1, x + 2,... ,x + s > with x (mod N) such that (x + j, N) = (M − j − (x + j), N) = 1, j ∈{1,... ,s}.

One can see that θs(M, N) = 0ifs is greater or equal than any prime factor of N; otherwise it is given by s + λ(p) θs(M, N) = N 1 − . (20) p|N p (1) When s = 1, θ (M, N) = θ1(M, N) = θ(M, N) is Nagell’s totient [181]. When (s) N | M, θ (M, N) and θs(M, N) reduce to Jordan’s totient Js(N) and Schemmel’s totient Ss(N) [181], respectively. Obviously, µ (M, N) is multiplicative in N. Other properties include: s N µs(M, d) · M, = e(n), (21) d|N d where (M, N) = 2 (M), m being the greatest divisor of N, relatively prime to M,and (m) = total number of prime factors of m; N θs(M, N) = dµs M, ; (22) d|N d (s) s N θ (M, N) = d µs M, ; (23) d|N d N µ(M, N) = µ(γ (d))µ , (24) d|N d (d,M)=1 where γ(N) denotes the largest squareefree divisor of N (”core”of N). In the paper there appear many other interesting identities, e.g. N ϕ(N) θs(M, d)µ = θs(M, N) ; (25) d|N d N 2 N θ (s)(M, N) θ (s)(M, d)µ M, = . s (26) d|N d N

~~133 CHAPTER 2~~

4 Cohen’s Mobius¨ functions and totients The extension of Euler’s totient (as in the above paragraph) leads many times to generalized Mobius¨ functions. In 1963 E. Cohen [51] deﬁned such an extension, involving a Mobius¨ generalization. Let p1, p2,... denote the increasing sequence of primes, and for a positive integer n, let νi (n) denote the multiplicity of pi as a divisor of n for each i ≥ 1. In particular, ν1(n) = 0ifpi n.Given a positive integer k, n is called k-free if νi (n)

~~The following elementary properties follow: µ ( ) ϕ ( ) = r d ; r n n (29) d|n d~~

~~ r γ (n), if n ∈ L + µ (d) = r 1 . (30) r 0, othwewise d|n where γ is the core of n. An interesting property follows from the deﬁnition:~~

~~ϕ ( ) = ϕ( ). lim r n n (31) r→∞~~

~~The following asymptotic result is true:~~

~~ax2 ϕ(n) = + O(x log2 x) (32) n≤x 2~~

~~134 THE MOBIUS¨ FUNCTION~~

ϕ( ) = ϕ ( ) ( n 1 n ), a x ϕ ( ) = r + ( 1+ε), r n O x (33) n≤x 2 with a, ar constants. More generally, one can introduce Mobius¨ functions of arbitrary direct factor sets. Let P, Q =∅be subsets of N such that if n1, n2 ∈ N, (n1, n2) = 1, then n = n1n2 ∈ P ⇔ n1, n2 ∈ P; and similarly for Q.Ifevery integer n ∈ N possesses a unique factorization of the form n = ab (a ∈ P, b ∈ Q), then each of the sets P and Q will be called a direct factor set of N.Inwhat follows P will denote such a direct factor set with (conjugate) factor set Q. The Mobius¨ function can be generalized to arbitrary direct factor sets, by writing

n µP (n) = µ , (34) d|n,d∈P d where µ is the classical Mobius¨ function. Forexample, µ1(n) = µ{1}(n) and µN(n) = δ(n) (where δ(n) = 1, n = 1; 0, n > 1). It is immediate that for n = n1n2, (n1, n2) = 1, n1, n2 ∈ P one has µP (n) = µP (n1)µP (n2).Onthe other hand:

~~n µP = δ(n) (35) d|n,d∈Q d which leads to the following Mobius-type¨ inversion formula:~~

~~n n f (n) = g ⇔ g(n) = f (d)µP (36) d|n,d∈Q d d|n d~~

~~By using these notions and properties, generalized totient functions may be introduced (see also E. Cohen [52]).~~

~~5 Klee’s Mobius¨ function and totient~~

V. L. Klee [124] in 1948 introduced the function Tk(r) as the number of integers h among the numbers 1, 2,... ,r for which (h, r) is k-free. For k = 1, Tk ≡ ϕ.Klee deﬁned a generalized Mobius¨ function µk by 1 µ r k , if r is a kth power µk(r) = . (37) 0, otherwise

~~135 CHAPTER 2~~

~~Then the function Tk can be expressed as r Tk(r) = µk(d) , (38) d|r d i.e.~~

~~Tk = µk ∗ E.~~

For ﬁxed r, T2(r) represents the number of those arithmetic progressions rs + t (t ∈{0, 1, 2,... ,(s − 1)}) which contain inﬁnitely many squarefree numbers. E. T (r) K. Haviland [104] showed that 2 is a Bohr almost-periodic function with an ab- r solutely and uniformly convergent Fourier series. A generalization of Klee’s function Tk has been introduced by D. Suryanarayana [53]. This is based on a notion of k-ary divisor of an integer. Let the symbol (a, b)k denote the greatest among the common k-th power divisors of a and b.If(a, b)k = 1 we say that a is relatively k-prime to b and vice-versa. A divisor d of n is called n k-ary if d, = 1. We note that for k = 1 one reobtains the unitary divisors. Let d k Tk(x, n) denote the number of positive integers ≤ x which are relatively k-prime to n. Then Tk(n, n) = Tk.Asageneralization of (38) one has x Tk(x, n) = µk(d) (39) d|n d

6Mobius¨ functions of Subbarao and Harris; Tanaka; and Venkataraman and Sivaramakrishnan In 1966 M. V. Subbarao and V. C. Harris [192] introduced the multiplicative function µk,q (r) by   1, if a ≡ 0 (mod k) µ ( a) = − , ≡ ( ) . k,q p  1 if a q mod k (40) 0, otherwise They proved the interesting relation

~~lim µk,1(r) = µ(r) (41) k→∞ and that, if ϕk,q (r) denotes the number of integers a (mod r) such that (a, r) has its kth power free part qth powerfree, then r ϕk,q (r) = µk,q (d) . (42) d|r d~~

~~136 THE MOBIUS¨ FUNCTION~~

~~From (42) follows also that~~

lim ϕk,1(r) = ϕ(r). (43) k→∞ Many generalized totient functions are also surveyed in [165]. Let λ be the Liouville function, given by λ(n) = (−1) (n) (where (n) denotes the total number of prime factors of n). A common generalization of Liouville’s function and the Mobius¨ function has been introduced by M. Tanaka [202]. Let k ≥ 2 be positive integer or inﬁnite and deﬁne   λ(n), if n is k − free µ ( ) = , k n  0 otherwise (44) λ(n), if k =∞

Clearly, µ2(n) = µ(n), µ∞(n) = λ(n). For 2 ≤ k ≤∞let Mk(x) = µk(x). Then n≤x √ √ Mk(x) − Bk x = ±( x) (45) 1 (for the symbol ± see Handbook I [142]) where B = 0, B∞ = , B = 2 ζ(1/2) k ζ(k/2)/ζ(1/2)ζ(k),ifk is odd; and Bk = 1/ζ(1/2)ζ(k/2) if k is even. Another generalization of the Mobius¨ function, leading to an extension of Ra- manujan sums has been introduced by C. S. Venkataraman and R. Sivaramakrishnan [212]. Put 0, if n contains a squared factor > 1, µ (n) = (46) k exp(πiω(n)/k), otherwise where ω(n) denotes the number of distinct prime factors of n and k ≥ 1. Clearly ω(n) µ1(n) = µ(n) by exp(πiω(n)) = (−1) . One has 1, if n = 1 µ (d) = (47) k (1 + s)ω(n), if n > 1 d|n where s = exp(πi/k). The function µk leads to a parallel generalization of Euler’s totient, as well a Ramanujan sums, by introducing

~~n ϕk(n) = µk · d, (48) d|n d~~

~~137 CHAPTER 2 and~~

~~n Ck(n, r) = µk · d (49) d|(n,r) d~~

The functions ϕk and Ck are multiplicative (the second one is both n and r), and are connected by the following formula: t Ck(n, r) = µk(r/g)ϕk(g) (50) ϕk(t) where g = (n, r) and t is the product of the common distinct prime factors of r/g and g.

7Mobius¨ functions as coefﬁcients of the cyclotomic polynomial In 1983 R. Dussaud [165] introduced a generalized Mobius¨ function, by application of results a cyclotomic polynomial Φn(x),having as roots the primitive roots of unity of order n. It is well known that the sum of all primitive roots of unity of order n is µ(n). Therefore, write

~~2 3 Φn(x) = 1 − µ(n)x + µ2(n)x − µ3(n)x + ... (51)~~

So µ2(n) is the sum of all zeros of Φn taken two-by-two, i.e. µ2(n) = xi x j (52) i< j where (xi )1≤i≤ϕ(n) are the roots of Φn(x) = 0. In an analogous way, Dussaud deﬁnes µk(n) by µ ( ) = ··· . k n xi1 xi2 xik (53) i1

~~138 THE MOBIUS¨ FUNCTION~~

We note that in condensed form, by another general method, these can be found in a paper by M. Deaconescu and J. Sandor´ [59]. In fact, this generalization involves the coefﬁcients of the cyclotomic polynomial and this question has been extensively studied (but not completely solved) by many authors (see e.g. [19], [159], [217]). Finally, we note that other generalizations of the Mobius¨ function in the spirit of this paragraph are due to Golubev (see [24]); H. Gupta (see [83]); C. A. Nicol (see [146]); M. H. Eggar (see [66]).

~~2.4 Mobius¨ functions of posets and lattices~~

1 Introduction, basic results The earliest papers on Mobius¨ functions and inversion theorems in settings other that of Number theory were concerned with lattices of subgroups of a group. L. Weis- ner [215], [216], P. Hall [85] were interested in prime-power groups, while S. De- lasarte [61] and E. Cohen [49], [50] in Abelian groups. M. Ward [213] considered more abstract cases. He defined the convolution of incidence functions of a locally finite lattice, and pointed out that such functions form a ring with respect to addition and convolution. See also a paper by R. Wiegandt [219] (who mentions interesting application of Delsarte’s inversion formula in group theory, due to L. Redei´ [156], [157]). For a recent book studying also the ”Wiegandt convolution”, see J. Golan [77]; and for related work on Incidence algebras, see E. Spiegel, Chr. J. O’Donnell [188]. In 1964, G.-C. Rota [161] laid the foundations of the theory of Mobius¨ function in combinatorial mathematics. His paper is filled with historical comments and examples (not recognizing, however the importance of the fundamental paper by R. Wie- gandt), and has an extensive bibliography. A number of applications of the Mobius¨ functions are contained e.g. in the paper by E. A. Bender and J. R. Goldman [11]. The important papers by D. Smith [184], [187] survey his results in this theory, while the work of H. Scheid [171] has an arithmetical emphasis. In what follows, only the most important notions and results of this theory will be mentioned. P =∅will be a partially ordered set (abreviated as poset), with ”≤”apartial ordering. For x, y ∈ P, the set

[x, y] ={z ∈ P | x ≤ z ≤ y} will be called an interval determined by x and y. P is called locally finite, if [x, y]isfinite for all x, y ∈ P.Inwhat follows, we shall always assume that P is locally finite.

~~139 CHAPTER 2~~

~~A function f : P × P −→ C such that f (x, y) = 0 whenever x ≤ y will be called an incidence function (or generalized arithmetical function). Let~~

~~F(P) ={f | f : P × P −→ C incidence function }.~~

The functions 1, x = y δ(x, y) = ; 0, x = y 1, x ≤ y e(x, y) = ; 0, x ≤ y clearly are in F(P). Define f + g and f · g naturally in F(P), and define also their convolution by ( f ∗ g)(x, y) = f (x, z)g(z, y), (x, y ∈ P), (1) x≤z≤y where a sum over an empty set is taken be 0. It can easily proved that (F(P), +, ∗) is a ring with unity element δ (however, this is not necessarily a commutative ring). This ring will be called as the incidence ring of F,orthe ring of generalized arithmetical functions defined on P. An incidence function f on P is said to have an inverse, if there exists g ∈ F(P) with

~~f ∗ g = g ∗ f = δ.~~

~~If there exists an inverse of f , this is unique, let in notation be f −1. As in the case of classical arithmetical functions, the following result holds:~~

~~f ∈ F(P) has an inverse iff f (x, x) = 0, ∀x ∈ P. (2)~~

~~The function e has an inverse. This is called the Mobius¨ function of P,denoted by µ. If f ∈ F(P), then~~

~~g = f ∗ e ⇐⇒ f = g ∗ µ; (3)~~

~~h = e ∗ f ⇐⇒ f = µ ∗ h. (4)~~

~~If x, y ∈ P, then we say that y covers x if x < y and if there is no element z ∈ P such that x < z < y.Inthis case~~

~~0 = δ(x, y) = (µ ∗ e)(x, y) = µ(x, x) + µ(x, y),~~

~~140 THE MOBIUS¨ FUNCTION hence~~

~~µ(x, y) =−µ(x, x) =−1. (5)~~

If x < y,buty does not cover x, there is nothing we can say generally on the values of µ(x, y).There is one case, however when can determine µ(x, y), that is the case of chains (or totally ordered sets), i.e. when for all x, y ∈ P one has x ≤ y or y ≤ x. If x, y, then the elements of [x, y] can be odered z0, z1,... ,zn so that

~~x = z0 < z1 < ...~~

~~For i = 0, 1,... ,n − 1, zi+1 covers zi ,hence µ(zi , zi+1) =−1. Assuming, that y does not cover x,wehaven ≥ 2. Then~~

µ(x, x) = 1,µ(x, z1) =−1,µ(x, z2) =−µ(x, x) − µ(x, z1) = 0, and by induction µ(x, zi ) = 0forall i ≥ 2. In particular, µ(x, y) = 0. The poset P is called a lattice if for all x, y ∈ P, x and y have a least upper bound (which is unique) denoted x ∨ y, and a greatest lower bound, denoted by x ∧ y. For standard and important properties of lattices, see e.g. [16], [201]. Certain examples are important to be noted here (see also [36]). On N, the relation m | n is a partial ordering. Let the resulting poset be denoted by ND.Ifm | n, then [m, n] ={d : d | n and m | d}, hence ND is locally ﬁnit. This is lattice, since x ∨ y = l.c.m.(x, y),

~~x ∧ y = g.c.d.(x, y)~~

~~(this lattice is even distributive). Let   n , | f m if m n ( , ) = , f m n  (6) 0, if m |n~~

~~Since 1 | n for all n, the mapping f −→ f is one to one, and furthermore preserves addition and convolution, i.e.~~

~~f + g = f + g,~~

~~f ∗ g = f ∗ g,~~

141 CHAPTER 2 where ”*” on the left is a Dirichlet convolution, while the right ”*” is the convolution in F(ND). If m | n, then d n ( f ∗ g)(m, n) = f (m, d)g(d, n) = f g = d|n d|n m d | | m d m d n n = f (e)g = ( f ∗ g) . | n me m e m Let now A be a regular convolution. The relation

m ≤ n if m ∈ A(n) (7) is a partial ordering on N. The resulting locally ﬁnit poset will be denoted by NA.If f is an incidence function of NA, then f is an incidence function of ND.Furthermore, if f, g ∈ F(NA), then their sum in F(NA) is the same as their sum in F(ND), and their convolution in F(NA) is the same as their convolution in F(ND). This is because the extra terms which appear in f (m, d)g(d, n) d|n m|d are actually equal to zero. Thus F(NA) can be considered to be a subring of F(ND). However the element e in F(NA) is not the same as the element in F(ND) (unless A = D)! With each arithmetical function f we can associate incidence functions f of NA deﬁned by   n , ∈ ( ) f m if m A n ( , ) = , f m n  0, if m ∈ A(n)

~~The mapping f −→ f is one to one and m ∈ A(n) implies~~

n n ( f + g)(m, n) = f (m, n) + g(m, n) = f + g = m m = ( + )( , ) = (+ )( , ), f g n m f g m n n ( f ∗ g)(m, n) = ( f ∗ g) = ( f∗ g)(m, n). A m A We note that (unless A = D), f ∈ F(NA) and f ∈ F(ND) are not generally the same.

~~142 THE MOBIUS¨ FUNCTION~~

The poset NA is not always a lattice. Since NA is locally finite and since 1 ≤ m and 1 ≤ n,every pair of elements of NA has a greatest lower bound in NA.However, α β if p is a prime and α = β, then p and p have no upper bounds at all in NU . If (P, ≤) is a locally finite partially ordered set, we can define the following generalized convolution (see [101]): ( f ∗K g)(x, y) = f (x, z)g(z, y)K (x, y; z), (8) x≤z≤y which is called K -convolution. If K ≡ 1, then we obtain the usual convolution (see [1]). If we have

~~K (x, y; x) = K (x, y; y) = 1whenever x ≤ y (9) then δ serves as the identity with respect to K convolution,~~

~~f ∗K δ = δ ∗K f = f, for all incidence function f . We remark that if~~

K (x, z2; z1)K (x, y; z2) = K (x, y; z1)K (z, y; z2) whenever x ≤ z1 ≤ z2 ≤ y, (10) then the K convolution is associative. It should be noted that the K convolution of incidence functions is a generalization of the K convolution of arithmetical functions (see (35) of 2.2).

~~2Factorable incidence functions, applications Let L be a local lattice. An incidence function f of L is called factorable if~~

~~f (a ∨ b, c ∨ d) = f (a, c) f (b, d), (11) for all a, b, c, d ∈ L such that a, b, c, d belong to the same interval in L and~~

~~c ∧ d ≤ a ≤ c, c ∧ d ≤ b ≤ d.~~

An example is in the case of ND.Anarithmetical function f is a multiplicative function iff the associated f ∈ F(ND) is factorable. The following results are true: (1) If L is a locally distributive, local lattice then the inverse of a factorable function f in F(L) is factorable.

~~143 CHAPTER 2~~

(2) If L is as above, then the convolution of two factorable functions in F(L) is factorable. (3) If L is a local lattice and µ is factorable, then L is locally distributive. The same if e ∗ eisfactorable. (4) Let f be a factorable function in F(L), constant on chains, and let h be a factorable companion of f (i.e. h(x, y) = f (x, y) − 1 whenever y covers x). If x, y, t ∈ L and x ≤ y ∧ t, then f (x, z)µ(z, y) = µ(x, y)µ(y ∧ t, y)h(y ∧ t, y) (12) x≤z≤y z∧t=x This important result is due to D. Smith [184]. Foranapplication, let f be a (classical) multiplicative function such that f (pα) = f (p) for all primes p and α ≥ 1. Let h be multiplicative function such that h(p) = f (p) − 1 for all primes. Then for all m, r one has the identity:

r f (d)µ = µ(r)µ(m)h(m), (13) d|r d (n,d)=1 r where m = . (n, r) We note that (13) could be called as the generalized Brauer-Rademacher identity, r 1 as for f (r) = , h(r) = we get ϕ(r) ϕ(r) µ( )µ( ) d µ r = r m , ϕ( ) ϕ( ) (14) d|r d d m (n,d)=1 which is the classical Brauer-Rademacher identity [90]. For another application, let A be regular arithmetic convolution and f be a multi- α t α plicative function such that f (p ) = f (p ) with t = tA(p ). Let h be multiplicative such that h(pα) = f (pα) − 1 for all primitive prime powers pα.Thenone has the identity

r f (d)µ = µ(r)µ(m)h(m); (15) A d d∈A(r) (n,d)A=1 r with µA being the Narkiewicz Mobius¨ function, and m = . (n, r)A In his series of papers [184]-[185] D. Smith has found a general context for a common study of Dirichlet, Cauchy, Lucas-Carlitz convolutions (see the preceeding paragraphs).

~~144 THE MOBIUS¨ FUNCTION~~

3Inversion theorems and applications We quote an inversion theorem, essentially due to L. Weisner [215]: 1) Let P be a poset, and f : P −→ C such that f (x) = 0 unless x ≥ t for some t ∈ P. If g(x) = f (z), (x ∈ P) z≤x then f (x) = g(z)µ(z, x), (16) z≤x and conversely. 2) Let P, f be as above, but f (x) = 0 unless x ≤ p for some p ∈ P. Then, if g(x) = f (z), z≤x then f (x) = µ(x, z)g(z), (17) z≤x and conversely. A parametric Mobius¨ inversion formula has been recently introduced By B. Chen [39]. Let P be a locally ﬁnite poset with incidence set F(P) and convolution (1). Let A be an arbitrary nonempty set and consider the vector space F(P × A) of all complex-valued functions on P × A. The incidence algebra F(P) acts naturally on F(P × A) in the left by D f ξ(x, a) = f ξ(x, a) = f (x, y)ξ(y, a), x≤y∈P where f ∈ F(P), ξ ∈ F(P × A). Similarly, for η ∈ F(A × P), the action in the right can be deﬁned by D f η(a, y) = η f (a, y) = η(a, x) f (x, y). x≤y,x∈P

~~Let A and B be nonempty sets. For any a ∈ A and b ∈ B, the following discrete analog of integration by parts is obvious: D f ξ(a, x)η(x, b) = ξ(a, x)D f η(x, b). x∈P x∈P~~

~~145 CHAPTER 2~~

We assume that all the posets are finite or countable infinite and that all the infinite sums are absolutely convergent, i.e. every infinite series converges to an element which is independent of the combinations of the terms in the sum. The following parametric inversion theorem holds true: For f ∈ F(P), ξ ∈ F(P × A),if f is invertible, and η(x, a) = f (x, y)ξ(y, a) (18) x≤y∈P is absolutely convergent, then ξ(x, a) = f −1(x, y)η(y, a) (19) x≤y∈P Particularly, if P = N, A = C (set of complex numbers), then we can state the following inversion theorem: Let f be a function on the lattice N × N,andlet ξ be a function on N × C.If f (k, k) = 0 for k ∈ N, and the series ∞ η(n, a) = f (n, kn)ξ(kn, a) (20) k=1 is absolutely convergent, then ∞ ξ(n, a) = f −1(n, kn)η(kn, a) (21) k=1 Some consequences of this inversion formula has applications in certain inverse problems in Physics. In 1990 Nan-Xian Chen [40] has shown the possibility of application of Mobius¨ inversion theory to physical problems. These are the inverse problems for obtaining the photon density of states, the inverse black-body radiation problem for remote sensing, and the solution for inverse Ewald summation. In 1993 Shao-jun Liu, Mi Li and Nan-Xian Chen [137] used Mobius¨ transform for inversion from cohesion to elastic constants, in 1995 M. Li and N.-X. Chen [135] gave a precise method to get the pair potential of diamond-type materials from their cohesive energy, while in 1998 N.-X. Chen and Rong Er-qian [41] found a concise and unified solution (based on Mobius¨ inversion) of inverse specific heat problems in physics.

4Mobius¨ functions on Eulerian posets We now study certain generalized Mobius¨ functions on Eulerian posets. A graded poset P is a ﬁnite partially ordered set with a unique minimum element 0, a unique maximum element 1, and a rank function ρ : P → N satisfying

~~146 THE MOBIUS¨ FUNCTION~~

ρ(0) = 0, ρ(y) − ρ(x) = 1 whenever y ∈ P covers x ∈ P. The rank ρ(P) of a graded poset P is the rank of its maximum element. Given a graded poset P of rank n + 1and a subset S ⊂{1, 2,...,n}, deﬁne the S-rank-selected subposet of P to be the poset PS ={x ∈ P : ρ(x) ∈ S}∪{0, 1} (22)

~~Denote by fS(P) the number of maximal chains of PS. Equivalently, fS(P) is the number of chains x1 < ···< x|S| in P such that~~

~~{ρ(x1), ...,ρ(x|S|)}=S.~~

~~The Mobius¨ function of a graded poset P is deﬁned recursively for any subinter- val of P by   1, ifx = y µ( , ) = [x y]  − µ([x, z]), otherwise (23) x≤z~~

~~Agraded poset P is Eulerian if~~

µ([x, y]) = (−1)ρ(x,y), (24) where ρ(x, y) = ρ(y) − ρ(x), for all [x, y] ⊂ P.For a survey of Eulerian posets, we quote R. P. Stanley [189]. In 2000 M. M. Bayer and G. Hetyei [7] have introduced the k-Mobius¨ function of a graded poset. This is deﬁned recursively by   1, if x = y 1 µk([x, y]) = (25)  −1 − µk([x, z]), otherwise k x

The following proposition gives the k-Mobius¨ function of a poset P as a k- analogue of the reduced Euler characteristic of the order complex of P. If P is a graded poset of rank n + 1, then |S| µ ( ) =− −1 n+1( ) k P fS P (26) S⊂{1,2,...,n} k

~~Agraded poset is k-Eulerian if for every interval [x, y] ⊂ P, one has~~

~~ρ(x,y) µk([x, y]) = (−1) (27)~~

~~147 CHAPTER 2~~

From the deﬁnitions it follows at once that if P is a k-Eulerian poset of rank n+1, then every interval of P is k-Eulerian; (28) and that n i−1 n (−1) fi (P) = k(1 − (−1) ) (29) i=1 For the existence of k-Eulerian posets, Bayer and Hetyei obtain the following characterization: Forevery positive integer n, there exists a k-Eulerian poset of rank n + 1ifand only if k = j/2 for some positive integer j. Moreover, every k-Eulerian poset is 2k-thick (30) (i.e. every nonempty open interval has at least 2k elements).

5 Miscellaneous results Forarecent paper on Mobius¨ function of lattices, see A. Blass and B. F. Sagan [18], where the concept of a bounded below set is introduced. This can be used to give a generalization of a theorem of Rota (on ”broken circuits” on a finite lattice). This result can be used to compute and combinatorially explain the Mobius¨ function in various examples including non-crossing set partitions, schuffle posets and integer partitions in dominance order. This paper also contains many References. In another paper, the homotopy and homology of finite lattices is considered, originated from the Mobius¨ function of geometric lattices [17]. For a recent paper containing a generalization of Mobius’¨ function arising in intuitionistic fuzzy theory (with application in number theory) see K. T. Atanassov [4]. For a study of Mobius¨ categories, and functors between such categories (containing a general setting for Mobius¨ inversion which includes the cases of locally finite partially ordered sets and monoids with finite decomposition property), see M. Content, F. Lemay and P. Leroux [54]. However, we do not consider here this very extensive field of study.

~~2.5 Mobius¨ functions of arithmetical semigroups, free groups, ﬁnite groups, algebraic number ﬁelds, and trace monoids~~

1Mobius¨ functions of arithmetical semigroups The theory of arithmetical functions, particularly Mobius’¨ function of an arithmetical semigroup was begun by A. Beurling in 1937 [13]. One can ﬁnd a good account of results in the book by J. Knopfmacker [125].

~~148 THE MOBIUS¨ FUNCTION~~

Afree commutative semigroup G with identity element 1G generated by a countable set P is called arithmetical if in addition there exists a real-valued norm mapping |·|on G such that i) |ab|=|a|·|b| for all a, b ∈ G ii) the total number NG (x) of elements n ∈ G of norm |n|≤x is finite for each x ∈ R. The elements of P (i.e. the generators of G) are called primes.Clearly, every element n = 1G in G has a unique (up to the order of factors) factorization of the form α α = 1 ... r n p1 pr (1) with pi distinct elements of P. Complex valued functions defined on an arithmetical semigroup are called arithmetical. The Mobius¨ function µ is defined by  G  1, if n = 1G µ ( ) = (− )r , = = ···= = G n  1 if a1 a2 ar 1in(1) (2) 0, otherwise Let MG (x) = µG (n) (3) 1≤x be the generalized Mertens function. Assume that the following asymptotic axiom holds true: Axiom. There exist positive constants C and δ, and a constant η with 0 ≤ η<δ such that δ η NG (x) = Cx + O(x ) as x →∞ (4)

It can be shown (see [125], Chapter 6) that if Axiom (4) is true in G, then the prime number theorem holds true in G.Infact, if (4) holds, then the prime number theorem, and the estimate δ MG (x) = o(x ) (5) are equivalent. It is interesting to note that this equivalence is not true in general arithmetical semigroups, see W. Zhang [222]. Relation (5) can be strengthened to the following: If G satisﬁes Axiom (4), then for every a > 0onehas

~~δ −a MG (x) = O(x (log x) ) (6) which is due to J. Knopfmacher [126].~~

~~149 CHAPTER 2~~

Various estimates on primes, or special arithmetic functions can be found in [125]. See also P. Haukkanen [91] for various generalizations. For some special arithmetic functions, (e.g. on k-free integers) see S.ˇ Porubsky[´ 152]. For asymptotic results on multiplicative functions defined on an arithmetical semigroup, see R. Warlimont [214]. For semigroups with unique factorization, and Mobius¨ type inversion theorems, see Z. Chen, Y. Shen and N. Chen [42]. In the classical case when P is a multiplicatively independent subset of (1, ∞) and G is the multiplicative semigroup generated by P,Beurling’s classical theorem states that if for N(x) = NG (x), N(x) = D · x + x · ε(x)(x →∞), where ε(x) = o(1), and if we want (prime number theorem) x P(x) ∼ (x →∞), log x in order to get this conclusion the condition 1 ε(x) = O (x →∞) (log x)3/2+ is sufficient (here a+ denotes any number strictly largen than a), and the condition 1 ε(x) = O (x →∞) (log x)3/2 is not sufficient. After the work on Beurling (1937), the main contributors in the field were B. Nyman [149], P. Malliavin [140] (who connected estimates on ε(x) and estimates on P(x) − lix), P. T. Bateman and H. G. Diamond [6], H. G. Diamond [62], H. Muller [144] (who looked for more refined conditions on ε(x) implying PNT (prime number theorem). Sz. Rev´ esz´ [158] went in another direction opened by Beurling: what can we say on P(x) when the assumption on N(x) is of the form N(x) = xQ(log x) + xε(x)(x →∞), Q(y) being either A · yρ (ρ > 0) or a more general perturbating factor. J.-P. Kahane used the approach of Bateman and Diamond, in [119] he proved a conjecture by them, namely

∞ dx (ε(x) log x)2 < ∞ 1 x implies PNT. For a Fourier formula, see [120], for a theorem of Littlewood for generalized primes see [121]. See also [122] for the role of Sobolev or Wiener algebras in the prime number theory of Beurling.

~~150 THE MOBIUS¨ FUNCTION~~

~~2Fee abelian groups and Mobius¨ functions~~

Let G be a free abelian group on a countable number of generators gi (i = 1, 2,...), and consider a homomorphism N : G → Q+ (positive rationals) such that N(gi ) are all integral. Therefore the images under N of all integral words of G are integers. Next we give a subgroup G of G which is of finite index h = [G : G], and denote a generic coset of G/G by H. Then the problem which arises is concerned with what can be deduced about the distribution (relative to N)ofgenerators of G in the cosets of G/G if one has information about the distribution of the integral words of G in these cosets. More precisely, letting w denote integral words, assume that for all cosets H of G/G one has δ BH (x) = 1 = CH · x + O(x ), (7) Nw≤x,w∈H 0 ≤ δ<1, CH ≥ 0, c = CH > 0 and aim at deriving for each H a result of the H form (prime number theorem) x x πH (x) = 1 = dH + o (8) Ng≤x,g∈H log x log x with cH , dH constants depending on H. Forexample, in the classical case of primes in arithmetic progressions the cH are all equal and the dH are all equal to 1/h; i.e. there is ”equi-distribution” of the generators of G over the cosets. In the case where the cH are unequal we may have in particular the other extreme wherein dG = 1 and for the cosets = G , dH = 0, i.e. 1 ”almost all” of the generators of G are in G.Ingeneral, for δ< , the situation is a 2 combination of these two. More precisely, consider the character group K of G/G and extend these characters naturally to characters of G; i.e. for any element t ∈ H, χ ∈ K we define χ(t) = χ(H).Then to each subgroup of K denote by #φ the set of elements t ∈ G such that χ(t) = 1 for all χ ∈ φ. Then #φ is a group and G ⊂ #φ ⊂ G.From the well known properties of the characters of finite abelian groups we then infer that the order of φ equals [G :#φ]. Next, introduce for each χ ∈ K A(χ) = cH χ(H) (9) H

~~When all the cH are equal, for χ = χ0 (principal character), one has A(χ) = 0. In other cases we may have χ = χ0 for which A(χ) = 0. The situation is saved,~~

~~151 CHAPTER 2 since for = (G/G) = the set of characters χ ∈ K with A(χ) = 0 can be proved (see W. Forman and H. N. Shapiro [71]) that~~

~~ is a subgroup of K (10)~~

~~Setting L = #, the following is true: If (7) holds with 0 <δ<1/2, then almost all generators of G are in L; furthermore,~~

~~the generators of G equi-distribute among the cosets of G in L. (11)~~

n m Thus if G = Lti , L = G s j are decompositions into disjoint cosets (t1 = i=1 j=1 1), G = G ti s j i=1,n, j=1,m is such a decomposition for G with respect to G.Theorem (11) then asserts that

~~1 x π (x) ∼ · (12) G s j m log x x whereas for the other cosets H, π (x) = o . Here H log x~~

~~h h m = [L : G] = = . [G : L] ord Thus, in this case   ord , ⊂ = if H L dH  h (13) 0, otherwise~~

1 If we do not assume δ< , the situation may be more complicated (see [71]). 2 An important step in the above theory is the asymptotic result that for all cosets H we have log Ng = D(H) log x + O(1) (14) nG≤x,g∈H Ng where D(H) is a ”Dirichlet density”. By using (14) and assuming (7) one can show that (8) holds with dH = D(H).

~~152 THE MOBIUS¨ FUNCTION~~

In the proofs a Mobius¨ function and a corresponding inversion formula constitute akey method. Let µG (w) be deﬁned by   1, if w = I (identity of G) µ (w) = (− )k, w = ... G  1 if g1 gk, where gi are distinct generators (15) 0, otherwise

Then it follows that 1,w= I µ (d) = (16) G 0, otherwise d|w where the summation is taken over all integral words d of G which ”divide” w, i.e. w = dd, with d an integral word of G. We then can state the following inversion formula: If P(w) is a complex-valued completely multiplicative function deﬁned over the integral words of G (i.e. P(ww) = P(w)P(w))with P(I ) = 1, and f (x) is a complex-valued function deﬁned for x > 0, then

~~x g(x) = P(w) f (x > ) ⇔ w 0 Nw≤x N~~

~~x f (x) = P(w)µ (w)g (x > ) G w 0 (17) Nw≤x N~~

Acharacter χ belongs to 1 if A(χ) = 0; to 2 if A(χ) = 0andL0(χ) = 0; and χ(w) to 3 if A(χ) = L0(χ) = 0. Here L0(χ) = lim . x→∞ w Nw≤x N Examples. 1) Let G = Q+. Fix any prime p,and any positive integer d, and let G be the subgroup of G generated by pd and the other primes. Then [G : G] = d, i and for the coset Hi = p G (i = 0, 1,...,d − 1),

~~pd−i−1(p − 1) 1 = + O(log x) (18) pd − 1 n≤x,n∈Hi~~

Here G/G is cyclic, and the coset H0 = G contains all primes other than p. Thus (G ) = K (G ) and the other sets 2,3 are empty. A(χ) = 0 for all characters, can be seen directly. Also, in this case L = G. 2) G = Q+, p afixed prime, d > 1fixed positive integer. Define G as the set of positive rationals of the form pαa/b where α ≡ 0 (mod d), (a, p) = (b, p) = 1,

153 CHAPTER 2 a ≡ b (mod p).HereG is a subgroup with [G : G] = d(p − 1) and cosets i Hij = p jG (i = 0, 1,...,d − 1; j = 1,...,p − 1). Also pd−i−1 1 = + O(log x) (19) pd − 1 n≤x,n∈Hij

In this case the primes are equi-distributed among the cosets H0 j ( j = 1, p − 1), 3 is empty, ord1 = d,andL is the subgroup composed of the elements in the cosets H0 j ( j = 1, p − 1). α = Q = π i ( ) = α = 3) G +.Forr pi let r i .Deﬁne G the set of r such that (r) ≡ 0 (mod 2). Hence [G : G] = 2, the cosets being G and 2G = H.Onthe assumption of the Riemann hypothesis, x 1 +ε 1 and 1 = + O(x 2 )(ε>0) (20) n≤x,n∈G n≤x,n∈H 2 In this case, all primes are in H, 2 =∅, ord1 = ord3 = 1, L = G = G. For equivalent formulations of abstract prime number theorems, see also P. Laborde Montaner and H. N. Shapiro [128].

3Mobius¨ functions of finite groups In his paper on Eulerian functions, P. Hall [86] in 1936 defined the Mobius¨ function on a subset of a power set, partially ordered by inclusion, and used the inversion formula to compute the number ϕs(G) of ordered s-tuples of elements which generate ∼ a finite group G.Heillustrated the method by evaluating ϕs(G) when G = PSL2(p) (p prime), and in the process computed the Mobius¨ value µ(G) for these groups. In 1959 W. Gaschutz¨ [73] by calculating ϕs(G) for soluble groups, implicitely obtained a formula for µ(G) in terms of the numbers of complements of chief factors. This formula was stated explicitly by Ch. Kratzer and J. Thevenaz´ [127]. In what follows, P will be a finite poset and µP will denote its Mobius¨ function. If P has a smallest element, say 0, we shall write µP (x) for µP (0, x). The µ(G) above referred is in fact µL(1, G), where L = L(G) is the poset of all subgroups of G. When disscussing subposets of L(G),weshall reserve the symbol µ for this meanning, too. Let G be a finite group, and denote by S(G) the poset of all subgroups of G. We will be mainly concerned with subposets P of S(G), and their Mobius¨ function µP .The notation µ refers always to µS(G) and µ(1, H) will be denoted simply by µ(H).IfL ≤ H ≤ G, then µ(L, H) can be calculated entirely within S(H) and is independent of the embedding of H in G;inparticular µS(G)(H) is an invariant of H.

~~154 THE MOBIUS¨ FUNCTION~~

~~H. H. Crapo [55] proved essentially the following result: Let N G and denote by K (G, N) the set of all subgroups of G which complement N. Then µ(G) = µ(G/N) µ(K, G). (21) K ∈K (G,N)~~

As a corollary, one can state that for a group G with µ(G) = 0, if N ≤ M G with G N G, one has that M/N is complemented in G/N.Inparticular ( ) = 1(for N Group theory, see e.g. [116]). Thus µ(G) = 0 whenever G has a Frattini chief factor. If N is an Abelian minimal normal subgroup of G with a complement K , then K is a maximal subgroup of G,soµ(K, G) =−1. We thus have another corollary: Let A be an Abelian minimal normal subgroup of G and let k denote the number of complements to A in G. Then

~~µ(G) =−kµ(G/A). (22)~~

~~This corollary gives an easy proof of a result by P. Hall [85]: Let X be a p-group of order pn. Then µ(X) = 0 unless X is elementary Abelian, in which case:~~

~~n (n) µ(X) = (−1) p 2 . (23)~~

~~But (22) applied inductively leads also to the Gaschutz¨ formula for the Mobius¨ value of a soluble group: Let~~

~~1 = H0 < H1 < ...< Hn = G be a chief series for the soluble group G, and let ki denote the number of complements to Hi /Hi−1 in G/Hi−1. Then~~

~~n µ(G) = (−1) k1k2 ···kn. (24)~~

The following general theorem is due to T. Hawkes, I. M. Isaacs and M. Ozaydin¨ [105]: Let G be a ﬁnite group whose order is divisible by n, and assume that n is divisible | | by G p, the order of a Sylow p-subgroup of G. If H is a subgroup whose order divides n, then [N(H) : H]p divides µ(H, X),where the sum is over all subgroups XofGwith |X| dividing n. When H = 1 and n =|G|p, this is exactly a classical theorem by K. S. Brown [22] from 1975.

~~155 CHAPTER 2~~

We now turn our attention to applications of Mobius¨ inversion formulae. For subgroups L and H of a group G, deﬁne αL (H, G) = µ(H, X) ⊆X where the sum runs over all subgroups X of G containing < H, L >.Note that

αG (H, G) = µ(H, G). By applying a classical Mobius¨ inversion formula for posets, the following result follows. Let L, H ≤ G. Then µ(H, G) = µ(X, G)αL (H, X) (25) where the sum runs over subgroups X of G containing < H, L >. The following divisibility result is a consequence of (25): Let p be a prime and H a subgroup of G. Set L = Op(G). Then

~~αL (H, G) is divisible by |NG (H) : H|p. (26)~~

~~Kratzer and Thevenaz´ proved essentially the following: Let H be a subgroup of G and let m be the squarefree part of |G : G H|.Then~~

~~|NG (H) : H| divides m · µ(H, G). (27)~~

This implies (see Hawkes-Isaacs-Ozaydin¨ [105]): Let m be the product of those prime numbers p for which G/Op(G) is elementary Abelian of order p or p2. Then |G| divides m · µ(G). p Furthermore, for any prime p, |O (G)|p divides µ(K, G) K where

~~K runs over all complements to Op(G) in G. (28)~~

~~Frobenius’ classical theorem states that if n is a divisor of the order of a ﬁnite group G, then the number of solutions to the equation~~

~~xn = 1inG is divisible by n. (29)~~

~~156 THE MOBIUS¨ FUNCTION~~

This can be proved elementarily by the following result of P. Hall [86]: If X is a group, let ϕs(X) be the number of s-tuples (x1,... ,xs) of a group elements xi such that X =< x1,... ,xs >.Then obviously S |G| = ϕs(X). x≤G

Set µ∗(X) = µ(X, G). Then µ∗ is the Mobius¨ function associated with the dual poset of S(G) and so µ(G) = µ∗(1). Applying the Mobius¨ inversion formula, we get: ∗ s ϕs(G) = µ (X) ·|X| . (30) X≤G

In particular, ϕ1(G) is zero unless G is cyclic, in which case ϕ1(G) = ϕ(|G|) the usual ϕ-function of Number theory. If H ≤ G, let [H] ={H g | g ∈ G}, the conjugacy class of H,andlet C(G) denote the set of all conjugacy classes of subgroups of G.Weview C(G) as a poset with partial ordering ≤ deﬁned by the rule [H] ≤ [L]ifH ≤ L g for some g ∈ G. In general, the poset C(G) is neither a meet -nor a join- semilattice, but it naturally admits Mobius¨ function µC(G), denoted by λG . Also we write λG (G) for λG ([G]). The following important result is true: If G is soluble, then

~~ µ(G) = λG (G) ·|G |. (31)~~

~~The above result by Hawkes, Isaacs and Ozadyn¨ has been generalized by H. Pahlings [150] as follows. If G is a soluble group and H a subgroup, then~~

~~ µ(H, G) = [NG (H) : H ∩ G ]λ(H, G) (32) where λ(H, G) is the notation for λ([H], [G]). Forexample, for the simple groups G = PSL(2, p) (p > 3aprime)one has~~

~~µ(H, G) = [NG (H) : H]λ(H, G) (33) for every subgroup H of G. Agroup A has the N-property with respect to a group B if, by deﬁnition~~

~~|Hom(B, NA(T )/T )|=1,~~

157 CHAPTER 2 for all T with 1 < T ≤ A (this holds for example, when B is a simple group with |B|≥|A|). The following result enables us to compute the Mobius¨ function of a direct product ([105]). If A is nontrivial and has the N − property with respect to B, then

~~µ(A × B) = µ(B) · (µ(A) − s), (34) where s is the number of epimorphisms from B onto A. Foranapplication, when A and B are non-isomorphic simple groups, then~~

~~µ(A × B) = µ(A) · µ(B). (35)~~

Relation (34) enables us to compute µ(G) when G is a direct product of simple groups: Let A be a non Abelian simple group and G = A× A×...× A, the direct product of n copies of A. Then µ(G) = 0ifµ(A) = 0, or else

~~n−1 µ(G) = µ(A)n · (1 − rt), (36) r=1 |Aut(A)| with t = . µ(A) Forexample, when G is a direct power of n copies of A5, then~~

µ(G) = (−60)n · 3 · 5 · ...· (2n − 1), (37) so one can see the interesting fact that there exists a group G and a prime p such that p divides µ(G) = 0, but doesn’t divide |Aut(G)|. For groups whose commutative subgroup is an abelian Hall π-subgroup, the following theorem is true (see M. Bianchi, A. Gillio Berta Mauri and L. Verardi [14]): Let G be (a ﬁnite) group, whose commutator subgroup G is an abelian Hall π- subgroup (all Sylow subgroups of which are elementary abelian). Let R be the lattice of those subgroups of G which are normal in G, and µR(G ) the corresponding Mobius¨ function. Then

~~ µ(G) = µ(G/G )|G |·µR(G ) (38)~~

As a corollary of (38) one obtains that if G is a group with G elementary abelian Sylow p-subgroup of order pk (p prime, k > 1), then G/G is of squarefree order. Further, for all H ≤ G let H < G.Then

~~µ(G) =±|G|·µ(G) (39)~~

~~158 THE MOBIUS¨ FUNCTION~~

~~The following result gives a condition for µ(G) to be a prime power: Let G be a group such that G is the elementary abelian Sylow p-subgroup. If G/G is of squarefree order, then~~

~~µ(G) = pk (40) for certain k ≥ 1. Generally (see [14]) µ(G) = pk if and only if |G|=ps, s ≤ k; the Sylow p-subgroup is elementary abelian and the other Sylow subgroups~~

~~are cyclic of prime order. (41)~~

~~In fact, for each prime p and k ≥ 1 there is a metabelian group G such that~~

~~µ(G) = pn. (42)~~

~~Similarly, if n = pkqh (n, q primes, k, h ≥ 1), then~~

~~there exists a metabelian group G such that µ(G) = n. (43)~~

~~Therefore for ω(n) ≤ 2, the Mobius¨ function µ is surjective. It can be proved that this is true for all n ≤ 100 (see [14]), however~~

~~this may be not true even for simple groups (see [15], (44) where for a such group G one has µ(G) = 0).~~

4Mobius¨ functions of algebraic number and function-fields Let K be an algebraic number field of degree k over the rational field. A generalized Mobius¨ function µK (a) defined over the integral ideals of K is defined by H. N. Shapiro [175] as follows:   , a =  1 if 0 (− )m, µ ( ) = 1 if a is squarefree and is the product K a  (45)  of m distinct prime ideals 0, otherwise α If one defines F(a) = µK (b), then F is multiplicative function and F(p ) = b|a 0, so that 1, for a = 0 F(a) = (46) 0, otherwise

~~159 CHAPTER 2~~

~~This enables us to introduce a generalized inversion formula as follows: If f (x) is a given complex-valued function deﬁned for all x > 0, and~~

~~x g(x) = f , (47) Na≤x Na then~~

~~x f (x) = µK (a)g . (48) Na≤x Na~~

This inversion formula gives a possibility to estimate various sums involving integral ideals of K , and to deduce Selberg’s identity in case of the algebraic number ﬁeld K (see [175]): log2 Np+ log Nplog Nq = 2x log x + O(x) (49) Np≤x Npq≤x

This is equivalent to log Np ∼ x (x →∞) (50) Np≤x i.e. the prime ideal theorem. We note that the proof of (49) uses also an important result from algebraic number theory, namely: 1 = αx + O(x1−1/k) (51) Na≤x where α is a constant depending only on the ﬁeld K ,and

+ 2r1 r2 πr2 Rh α = √ (52) w |D| where r1 = number of real conjugate ﬁelds of K , r2 = number of imaginary conjugate ﬁelds, w = number of roots of unity in K , h = number of ideal classes (see [218]). The function µK of Mobius¨ enables us to introduce a generalized vonMangoldt function, too:   0, if a = 0 a (a) = µ (b) log N = log Np, if a = pα (53) K K b  b|a 0, otherwise

~~160 THE MOBIUS¨ FUNCTION~~

~~The following identity is true (see [175]): 2 a a µK (b) log N = K (a) log Na + K (b)K (54) b|a b b|a b~~

Besides Shapiro’s proof, another elementary proof of the prime ideal theorem (in the form (50)) is due to C. Toubi [209], who follows a method due to A. Hildebrand [106]. If P =∅is a set of prime ideals p ∈ P,M.Skalba [182] deﬁnes   µ(a) if a is an integral ideal of K µ ( ) = ∈ , P a  not divisible by any p P (55) 0, otherwise

This is applied in the proof of generalized Euler identities. For Mobius¨ and Euler type functions on algebraic function ﬁelds, see e.g. the monograph by M. Rosen [160]. For generalizations of von Mangoldt’s function ,seeA. Ivic´ [117], [118]; and for a characterization, see P. Haukkanen [102]. For a unitary Mangoldt function, see [30].

~~5Trace monoids and Mobius¨ functions~~

Finally, we shall deal with Mobius¨ functions in trace monoids.Given a finite ∗ alphabet ,atrace monoid M is defined as the quotient / ≡I , where I ⊂ × is a symmetric and irreflexive relation, and ≡I is the smallest congruence extending the set of equations {ab = ba| (a, b) ∈ I }.The elements of M are called traces. These were introduced in 1969 in P. Cartier and D. Foata [37] in order to give a completely combinatorial proof of Mac Mahon’s Master theorem (see G. Lallement [129], ch. XI). They were studied further as a possible model of parallelism by interpretation of generators of the monoid as actions and their commutation as independence of actions (A. Mazurkiewicz [141]). A theory of trace languages was then developed, extending the properties of formal languages. For a survey of results, as well as open problems, see V. Diekert and G. Rozenberg [64]. The pair , I is called independence alphabet and can be represented by an undirected graph, with the set of nodes, and I the set of edges. In what follows, Z M ! will be the ring of formal power series over M with integer coefficients. The support of σ ∈ Z M ! is the set of all t ∈ M with σ(t) = 0. A polynomial is a formal power series having finite support.

~~161 CHAPTER 2~~

The Mobius¨ function of a trace monoid M is deﬁned by  n  (−1) , if t = a1 ...an, where all ai ∈ are different, µ ( ) = ( , ) ∈ = , M t  and ai a j I for all i j (56) 0, otherwise with the convention that µM (1) = 1. If ξM = t is the characteristic series of M, t∈M then one has

ξM µM = µM ξM = 1, (57) see [37]. The Mobius¨ function µM can be represented by several polynomials in Z. Consider the canonical morphism ψ: Z !→ Z M !.Letus say that a polynomial µ ∈ Z ! is a non-commuting lifting of the Mobius¨ function ∗ of M if ψ(µ) = µM and the restriction ψ : → M is injective on the support of µ. Such a function is calledunambiguous if its inverse ξ satisfies ξ(w) ∈{0, 1} for all w ∈ ∗ and ψ(ξ) = t.V.Diekert [129] proved that an unambiguous non- t∈M commuting lifting µ of the Mobius¨ function of M exists iff the graph , I admits atransitive orientation. Assume that I is a transitive orientation of , I and let be a linear extension of I over . The formal power series given by   (− )n, x = x x ...x , {x ,...,x }  1 if 1 2 n 1 n is a clique , µ( ) = of I and x is a lexicographically smallest string x  , (58)  in x with respect to 0, otherwise is an unambiguous lifting of µM (see [63]). C. Choffrut and M. Goldwurm [44] call µ as the unambiguous Mobius¨ function defined by I .In[44] the following result is proved: Let , I be an independence alphabet that admits a transitive orientation I . Then the unambiguous Mobius¨ function µ defined by I is given by µ = det(In − X), where X is a matrix and In the identity matrix of order n.InfactX is a minimal acceptor matrix of a lexicographic cross section defined by I .For details, see [44]. Forrelated results on Algebraic topology, see [20], [70], [153], [204], [18] and [17].

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~~173 CHAPTER 2~~

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~~175 CHAPTER 2~~

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~~176 THE MOBIUS¨ FUNCTION~~

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~~177 Chapter 3~~

~~THE MANY FACETS OF EULER’S TOTIENT~~

~~3.1 Introduction~~

Motivated by a generalization of Fermat’s divisibility theorem, in 1760 L. Euler [133] proved that aϕ(n) ≡ 1(modn) for (a, n) = 1,(1) where ϕ(n) denotes the number of positive integers r < n such that (r, n) = 1. Euler studied also other important properties of this function, e.g. the multiplicativity property, which means that ϕ(mn) = ϕ(m)ϕ(n) whenever (m, n) = 1. (2) It was C. F. Gauss (see L. E. Dickson [103]) who introduced the symbol ϕ(n) for Euler’s ϕ-function, making the convention ϕ(1) = 1. In 1879 J. J. Sylvester introduced the word ”totient”, while E. Prouhet (1845) proposed the name ”indicator” (see [103]). Sylvester deﬁned also the ”totatives” of n to be the integers r < n with (r, n) = 1. Euler’s totient is of major interest in number theory, as well as many other ﬁelds of mathematics. For example, the apparently simple result (1) allowed mathematicians to build a code which is almost impossible to break, even though the key is made public (see for instance R. Rivest, A. Shamir and L. Adleman [352]). Euler’s totient is connected to many notions and functions of various domains. Gauss proved the identity ϕ(d) = n (3) d|n

179 CHAPTER 3 which by the Mobius¨ inversion formula (see Chapter 2) gives µ(d) 1 ϕ(n) = n = n 1 − (p prime)(4) d|n d p|n p For inequalities and estimates connecting ϕ to the arithmetical functions d (number of divisors), σ (sum of divisors), ω (number of distinct prime divisors), (total number of prime divisors), Jk (Jordan’s function), ψ (Dedekind’s function), (von Mangoldt’s function), ϕ∗, d∗,σ∗ (unitary analogs of ϕ,d,σ), etc., see D. S. Mitri- novic,´ J. Sandor´ and B. Crstici [291] (which we shall call sometimes in what follows Handbook I). On the other hand, many other aspects of Euler’s totient were not considered in [291], for instance: congruence properties similar to (1) (and generalizations and extensions); other congruence properties of ϕ; identities (such as Kesava Menon or Brauer-Rademacher identities); generalizations and extensions of ϕ;various equations involving ϕ; the cyclotomic polynomial; matrices and determinants (as the Smith determinant); connections to congruences, etc. All these subjects will be treated in what follows. In this Introduction we want also to mention certain unfamiliar occurrences of Euler’s totient.

1 The infinitude of primes E. Kummer (see [27]) derived the following simple proof of the infinitude of primes: suppose p1 < p2 < ··· < pk form the (finite) set of primes. Then by applying the prime factorization, for n = p1 p2 ...pk we must have ϕ(n) = 1. On the other hand, one has ϕ(n) = (p1 − 1)...(pk − 1)>1, a contradiction. Another proof, based on the divisibility property n|ϕ(an − 1)(a, n > 1 integers)(5) is given in M. Deaconescu and J. Sandor´ [97] (see also J. Sandor´ [362]). Properties similar to (5) will be studied in the next paragraph.

~~2 Exact formulae for primes in terms of ϕ~~

There exist in the literature various exact formulae expressing the n-th prime pn, or the prime-counting function π(n) (i.e. the number of primes ≤ n). For example by Williams, Mina´c,ˇ Sierpinski or Gandhi, see R. Ribenboim [349] (pp. 180-186), or [350] (where one can ﬁnd also the following statement ”... c’est inutile, mais c’est joli...”). In 1992 K. Atanassov [16] has found the formula n π(n) = sg(k − 1 − ϕ(k)) (6) k=2

~~180 THE MANY FACETS OF EULER’S TOTIENT~~

0, if x = 0 where sg x = 1, if x = 0 He proved also that 2n pn = sg(n − π(k)) (7) k=0 0, if x ≤ 0 where sg x = 1, if x > 0 M. Vassilev [464] proved that n ϕ(k) π(n) = ( ) − 8 k=2 k 1 where [x] denotes the integer part of x.

3 Inﬁnite series and products involving ϕ, Pillai’s (Cesaro’s)` arithmetic functions Euler’s totient occurs in many identities for special functions or constants.In 1857 J. Liouville (see [103], p. 120) proved that

~~∞ xn x ϕ(n) = for |x| < 1.(9) − n ( − )2 n=1 1 x 1 x~~

~~More generally, E. Cesaro` (see [103], p. 127) showed that~~

~~∞ ∞ xn f (n) = xn f (n), (10) − n n=1 1 x n=1 if F(n) = f (d). See also G. Polya´ and G. Szego¨ [332] (vol. II, Part 8, paragraph d|n 7). In [332] one can ﬁnd also identities like~~

~~∞ xn 1 + x2 ϕ(n) = x (due also to J. Liouville), (11) + n ( − 2)2 n=1 1 x 1 x~~

~~∞ xn 1 + x J (n) = x ,(12) 2 − n ( − )3 n=1 1 x 1 x ∞ xn 1 + 2x + 6x2 + 2x3 + x4 J (n) = x (13) 2 + n ( − 2)3 n=1 1 x 1 x~~

~~181 CHAPTER 3~~

where J2 is the Jordan totient function Jk for k = 2, and |x| < 1. Similarly, ∞ − n ϕ(n)/n x (1 − x ) = e 1−x (14) n=1 while the inﬁnite product ∞ n ϕ(n)/n 1 + x −1 = e2/(x −x) (15) − n n=1,odd 1 x has been proposed by P. J. Forrester and M. L. Glasser [136] (in fact (15) follows from (14) on replacing x by −x). A Mercier [289] generalized (15) as follows:

∞ ∞ ( )/ 1 + xn f n n 2 F(n)xn/n = e n=1,odd (16) − n n=1,odd 1 x where F(n) is given in (10). An interesting inﬁnite product identity, involving also Fibonacci and Lucas numbers Fn and Ln is due to D. Redmond [348]: ∞ ϕ( )/ L n n √ √ √ n = e−(1+ 5)/2 5 (17) · n=1 5 Fn For ﬁnite arithmetical products, involving e.g. the gamma-function, too, see [384]. The constant ∞ ∞ 1 1 = 1 + = 1.78657 ... (18) ϕ( )σ ( ) 2k − k−1 n=1 n n pprime k=1 p p is called also as the ”Silverman constant” (see D. Rusin [357]). For other constants related to the Euler totient function, we quote S. Finch [135]. In the same paper in which he proved (9), Cesaro` considered also the sum of (n, i) for i ≤ n, where (n, i) is the g.c.d. of n and i: n n (n, i) = dϕ (19) i=1 d|n d The left side of (19) is called often as the Pillai arithmetic function P(n), who in 1933 [330] studied certain properties (without reference to Cesaro)` of this function. n n P(n) For three extensions of , see R. Sivaramakrishnan [420]. The function ( , ) i=1 i n occurs also in group theory, and for a common generalization of this function and Pillai’s see J. Sandor´ and A.-V. Kramer´ [382]. A generalization involving regular

182 THE MANY FACETS OF EULER’S TOTIENT convolutions of Narkiewicz (see Chapter 2) is due to L. Toth´ [455] (see [456] for a unitary analogue of P(n), along with asymptotic evaluations). See also [459]. For n other Cesaro` type functions, and particularly for the study of [i, n] (where [i, n] = i=1 lcm(i, n))see H. W. Gould and T. Shonhiwa [153], [154].

4 Enumeration problems on congruences, directed graphs, magic squares 2d − 1 n If one replaces d with in the ﬁrst term of the product dϕ in re- d d lation (19), one obtains the solution of an enumeration problem for a congruence equation. More precisely, R. A. Brualdi and M. Newman [56] have shown that the number of n-tuples (x0, x1,...,xn−1) of nonnegative integers such that

n−1 n−1 xi = n, ixi ≡ 0 (mod n)(20) i=0 i=0 is given by 1 2d − 1 n ϕ (21) n d|n d d In fact, this result follows also from a theorem by K. G. Ramanathan [344]. The number (21) gives also the number of formally distinct diagonal products of an n by n circulant matrix (see [56]). In 1994 A. Ehrlich [107] deﬁned for two positive integer a and n the directed graph G(a, n) with vertex set {0, 1,...,n − 1} such that there is an arc from x to y if and only if y ≡ ax (mod n). Let C(a, n) denote the number of cycles in G(a, n). Erlich proved that C(2, n) is odd or even according as 2 is or not a quadratic residue mod n. (22) E. Brown [54] has considered the general case, by proving that for C(a, n) one has the formula ϕ(d) C(a, n) = ( ) ( ) 23 d|n ordd a r where ordd (a) is the least positive integer r such that a ≡ 1 (mod d).This implies e.g. that for (a, n) = 1, n odd, one has a 1 + C(a, n) ≡ n (mod 2)(24) 2 a where is the Legendre-Jacobi symbol. n

~~183 CHAPTER 3~~

For n even, n = 2kn (n odd) C(a, n) is even, for k = 1; and −1 1 − n C(a, n) ≡ (mod 2)(25) 2 A generalization of Euler’s totient is the Schemmel totient function (see [103]), deﬁned by S2(n) = card{(x, x + 1) :0< x,(x, n) = (x + 1, n) = 1, x ≤ n}.It can be proved that 2 S2(n) = n 1 − (26) p|n p D. N. Lehmer [244] used Schemmel’s totient to the enumeration problem of certain magic squares.Many other generalizations of Euler’s totient will be studied in the next paragraphs of this Chapter.

5Fourier coefficients of even functions (mod n) Let n be a fixed positive integer, and let F be a field of characteristic 0 containing the nth roots of unity. In 1952 E. Cohen [76] defined a function f : N → F to be (n, F)-arithmetic,if f (a) = f (b) whenever a ≡ b (mod n).Anexample of such a function is f (a) = exp(2πija/n) which is (n, F)-arithmetic for any integer j. Let An(F) denote the set of all (n, F)-arithmetic functions. If f ∈ An(F),then f is called even (mod n) if f (a) = f ((a, n)), where (a, n) denotes the greatest common divisor of a and n.Taking F = C, the field of complex numbers, let Bn(C) denote the set of even functions (mod n). Let C(a, d) be the Ramanujan sum, defined by C(a, d) = e(ax, d)(27) (x,d)=1, 1≤x≤d with e(a, b) = e2πia/b.Infact C(a, d) can be defined for all integers a, and positive integers d, leading to a common generalization of Euler’s totient ϕ,andMobius’¨ function µ. Indeed, it is immediate that C(0, d) = ϕ(d), C(1, d) = µ(d) for all d ≥ 1 (28) E. Cohen [77] proved that if f ∈ B (C), then f can be written uniquely as n f (a) = α(d)C(a, d), (29) d|n where the coefficients α(d), d|n are called as the Fourier coefficients of f . These coefficients may be explicitly written (see P. J. McCarthy [382]) as 1 α(d) = f (a)C(a, n)() ϕ( ) 30 n d a (mod n)

~~184 THE MANY FACETS OF EULER’S TOTIENT~~

For f, g ∈ Br (C) let us introduce the inner product n n f, g = ϕ(d) f g = (ϕ ∗ f g)(n)(31) d|n d d where g(a) = g(a) is the complex conjugate of g(a), and ∗ is the Dirichlet convolution. It is known (see J. Knopfmacher [229]) that Bn(C), ·, · ) is a complex Hilbert space, with norm 1/2 f = n |α(d)|2ϕ(d) (32) d|n

P. Haukkanen and R. Sivaramakrishnan [189] have introduced a subspace Sq (C) of Bn(C) as follows. Let f ∈ Bn(C) and q be a ﬁxed divisor of n.Associated with f , one can construct an even function f (a, q) (even (mod n))bydeﬁning f (a, q) = α(d)C(a, d)(33) d|q

If d|n, d q, then α(d) = 0inthe representation of f (a, q) as an element of Bn(C) (therefore we have actually a truncated sum). Let Sq (C) be the set of functions of form (33). Then Sq (C) becomes a subspace of Bn(C),ofdimension d(q), where d is the number-of-divisor function. By the Riesz representation theorem one can write

(C) = (C) ⊕ ⊥(C)() Br Sq Sq 34 ⊥(C) (C) ( ) − ( ) where Sq is the orthogonal complement of Sq (of dimension d n d q ). The set 1 C(·, d) : d|n (35) (nϕ(d))1/2 forms an orthonormal basis of the Hilbert space Bn(C) (see [189]). For a measure theoretic approach, as well as linear transformations, see [189]. For multiple trigonometric sums, a similar study has been done in [190].

6 Algebraic independence of arithmetic functions In 1948 R. Bellman and H. N. Shapiro [36] have introduced a notion of algebraic independence of real valued arithmetic functions. A set of arithmetic functions fi (i = 1, N), fi : N → R are called algebraic independent over R if there exists no N polynomial P(X1,...,X N ) ≡ 0 with real coefﬁcients, which is irreducible over R such that P( f1(n),..., fN (n)) = 0, for all n ∈ N (36)

~~185 CHAPTER 3~~

Bellman and Shapiro obtain various general theorems, for example: Let E : N → R be the identity function; i.e. E(n) = n for all n; and let f be a multiplicative function. If E and f are algebraically dependent, then f (n) = nr , n ∈ N (where r is aﬁxedrational number). (37) Let fk+1(n) = fk(d), k = 0, N, where f0 is multiplicative, and suppose d|n that f0(p) takes on an inﬁnite set of values over some set of primes p ∈ P. Then f0,..., fN are algebraically independent. (38) As a corollary one gets that the functions

~~E,ϕ,σ are algebraically independent (39)~~

~~Another result says that~~

~~E,ϕ,σ,d, d∗ are algebraically independent (40) as well as: E,ϕ,σ,d, d∗,µare algebraically independent (41)~~

~~7 Algebraic and analytic application of totients In what follows, in this Introduction we mention an algebraic and an analytic application of totients. Jordan’s totient Jk is deﬁned by~~

~~Jk(n) = card{(a1,...,ak) :1≤ a1, a2,...,ak ≤ n,(a1,...,ak, n) = 1} (42)~~

Clearly J1 ≡ ϕ. This function has applications in the theory of linear groups (see C. Jordan [216]). For some recent applications in group theory, see J. Schulte [400]. R. Guitart [167] has applied this function to explicit computation of internal exponentiation of isomorphic classes of objects in the topos of permutations. See also [168]. The function Jk should not be confused with Jk,given by ( ) = {( ,..., ) ≤ ≤ ≤ ···≤ ≤ ,( ,..., , ) = } Jk n card a1 ak :1 a1 a2 ak n a1 ak n 1 ( ) 42 n k While Jk(n) = µ(d) , one has on the other hand, d|n d + − ( ) = µ n d k 1 Jk n d|n d k

~~186 THE MANY FACETS OF EULER’S TOTIENT~~

m( ) ≤ ,..., ≤ For the more general functions Jk n , where in (42) 1 a1 ak n and ( ,..., , ) = m( ) ≤ ≤ ··· ≤ ≤ a1 ak m 1; and Jk n where in (42’), 1 a1 ak n, ( ,..., , ) = n( ) = ( ) and a1 ak m 1, see T. Shonhiwa [412]. Remark that Jk n Jk n and n( ) = ( ) ( ) = {( ,..., ) Jk n Jk n . Shonhiwa studies also the functions Jk n card a1 ak : 1 ≤ a1 < a2 < ··· < ak ≤ n,(a1,...,ak, n) = 1}, and the corresponding function m( ) Jk n .For example, one has ( ) = µ n d Jk n d|n d k

The sum-of-divisors function σ is strongly related to Euler’s totient (see [291]). Now the following inequality is equivalent to the famous Riemann Hypothesis on the zeta function: The Riemann Hypothesis is true if and only if, for sufﬁciently large n one has

~~σ(n)~~

(Here γ is Euler’s constant). This is due to G. Robin [354]. 1 Recently A. Verjovsky [470] proved the following result: Let ≤ a ≤ 1. Then 2 the Riemann zeta function has no zeros in the half-plane Re s > a if and only if ∞ n a+ε ϕ(n) f = m0( f, x) + O(x ), ε > 0,(44) n=1 x for any smooth function f : R+ → R with compact support, where 6 ∞ m ( f, x) = x2 tf(t)dt. 0 2 π 0

~~8 ϕ-convergence of Schoenberg~~

~~In 1959 I. J. Schoenberg [398] deﬁned the ϕ-convergence of a sequence {γn} of real numbers as follows: ϕ − limit γn = λ if and only if lim sn = λ,where n→∞ 1 sn = ϕ(d)γd (45) n d|n~~

~~A sequence which is ϕ-convergent, but not convergent is given e.g. by , = · · ... γ = 1 if n 2 3 5 pk n 0, otherwise~~

~~187 CHAPTER 3~~

where pk is the kth prime (see the solution of Problem 6090 [361]). Schoenberg proved that if {nk} is a given sequence of increasing natural numbers, then the assumption ϕ − limit γn = λ implies that lim γn = λ if and only if k→∞ k

lim inf ϕ(nk)/nk > 0 (46) k→∞ From the existence of an asymptotic distribution function for ϕ(n)/n,itfollows that if the sequence (nk) has the property that lim ϕ(nk)/nk = 0, then the density of k→∞ (nk) is 0. Write d−lim γn = λ if d{n : |γn −λ|≥ε}=0 for every ε>0, where d denotes (asymptotic) density. Now, Schoenberg proves that if ϕ −lim γn = λ, then one has also d−lim γn = λ. (47) 1, n = prime The converse is evidently false (put γ = ). n 0, otherwise A criterion for d − lim γn = λ is contained in the following necessary and sufﬁ- cient condition: 1 γ γ γ λ lim (eit 1 + eit 2 + ···+eit n ) = eit (48) n→∞ n for every real t (see [398]).

~~3.2 Congruence properties of Euler’s totient and related functions~~

~~1 Euler’s divisibility theorem The most famous congruence property of Euler’s totient is clearly Euler’s divisibility theorem (see also paragraph 3.1):~~

~~aϕ(n) ≡ 1 (mod n) for (a, n) = 1 (1)~~

For the early history of this theorem, one can ﬁnd in Dickson’s book [103], that in 1776 P. S. Laplace obtained a proof different from Euler’s. In 1831 V. Bouniakowsky gave a proof similar to that by Laplace, while in 1832 Giovanni de Paoli gave another proof. In 1845 L. Poinsot proved Euler’s theorem by geometrical considerations: Let a be prime to n and < n. Join any vertex of a regular polygon of n sides with the ath vertex following it, the new vertex with the ath vertex following it, etc., thus deﬁning aregular star polygon of n sides. With the same a, derive similarly a new n-gon, etc., until the initial polygon is reached. The number k of distinct polygons thus obtained is seen to be a divisor of ϕ(n), the number of polygons corresponding to the various

188 THE MANY FACETS OF EULER’S TOTIENT a’s.Ifinthe initial polygon we take the akth vertex following any one, etc., we obtain the initial polygon. Hence ak,soaϕ(n) has the remainder unity when divided by n. In 1896 H. Weber deduced the ﬁrst proof of (1) based on Group theoretical considerations. In 1913 G. Frattini noted that if F = F(a, b,...)is a homogeneous symmetric polynomial, of degree g with integral coefﬁcients, in the integers a, b,... less than n and prime to n, and if F is prime to n, then

~~kg ≡ 1 (mod n) for any integer k,(k, n) = 1 (2)~~

In fact, F(a, b,...) ≡ F(ka, kb,...) ≡ kg F(a, b,...) (mod n).SoifF = ab ...,wereobtain Euler’s theorem. There is an interest, even today in the new proofs of Euler’s theorem. See H. Alzer [8] (where, in fact Redei’s´ theorem (11) is proved). K. Heinrich and P. Horak [197] obtain a combinatorial proof of (1) when m is a prime power. The proof of the general case follows from this fact by usual methods. For combinatorial and geometrical investigations of congruence theorems, see also K. Hartig¨ and J. Suranyi´ [180]. R. T. Hansen [176] formulated (1) for general a and n as follows: Let (a, n) = d. n Then aϕ(n)+1 ≡ a (mod n) iff d, = 1. (1) d 2 Carmichael’s function, maximal generalization of Fermat’s theorem It follows from Euler’s theorem that if (a, n) = 1, there exists the smallest positive exponent r such that ar ≡ 1 (mod n), (3) called the order of a modulo n, and denoted r = ordn(a). It is immediate that am ≡ 1 (mod n) if and only if m is a multiple of r; thus in particular, r|ϕ(n). It is natural to ask: given n > 2, does there always exist an integer a, (a, n) = 1 such that ordn(a) = ϕ(n)? When n = p is a prime, such numbers exist, namely the primitive roots modp.Ifn = pk, p odd prime, it is also true. However, if n is divisible by 4p,orpq, where p, q are distinct odd primes, then there is no number a, (a, n) = 1 such that ordn(a) = ϕ(n). Indeed, in 1910 R. D. Carmichael [65] (see also [66]) introduced a function λ,deﬁned as follows:

~~λ(1) = 1,λ(2) = 1,λ(4) = 2,λ(2a) = 2a−2 (r ≥ 3),~~

~~λ(pa) = ϕ(pa) = pa−1(p − 1)(r ≥ 1, p odd prime);~~

~~λ( a a1 a2 ... as ) = λ( a), λ( a1 ),...,λ( as ) ( ) 2 p1 p2 ps [ 2 p1 ps ] 4 where [...] denotes the least common multiple.~~

~~189 CHAPTER 3~~

~~Note that λ(n)|ϕ(n)(5) so λ(n) ≤ ϕ(n),butλ(n) = ϕ(n) only for n = 1, 2, 4, pa, 2pa.Carmichael proved the following divisibility theorem:~~

~~aλ(n) ≡ 1 (mod n) for (a, n) = 1 (6)~~

~~As a corollary of (6) one obtains that, except for the cases n = 1, 2, 4, pa or 2pa, we have 1 ϕ(n) a 2 ≡ 1 (mod n)(6 )~~

He also showed that there always exists an element a such that ordn(a) = λ(n). One can ﬁnd in [103] that E. Lucas (1890) stated, without proof the following result: = a1 a2 ... ak Let n p1 p2 pk be the prime factorization of n and put

~~H(n) = max{a1, a2,...,ak}.(7) Then for all positive integers a and n~~

~~aλ(n)+H(n) ≡ a H(n) (mod n)(8)~~

This is a maximal generalization of Fermat’s theorem written in the form a p ≡ a (mod p) for any prime p,any integer a.For a proof of (8), see D. Singmaster [419]. The fact that (8) is maximal generalization, follows also from the following remark (see [419]): Let r > s. Then ar ≡ as (mod n) for all a and n iff λ(n)|(r − s) and s ≥ H(n).(9) The following inequalities are true:

~~λ(n) + H(n) ≤ ϕ(n) + H(n) ≤ n (10) with simultaneous equality iff n is prime, or n = 4. From (8), (9), (10) the following corollaries can be stated:~~

~~an ≡ an−ϕ(n) ≡ an−λ(n) (mod n); (11)~~

aϕ(n)+H(n) ≡ a H(n) (mod n), for any a and n.(12) We note that the ﬁrst congruence of (11) is attributed to L. Redei´ (see T. Szele [451], S. Schwarz [404]), while (12) is mentioned in N. Nielsen [321]. It is interesting to note also that Carmichael’s theorem (6) has applications in the theory of the so-called ”3x + 1 problem”(afamous unsolved problem), see J. C. Lagarias [234] and E. Belaga and M. Mignotte [35].

~~190 THE MANY FACETS OF EULER’S TOTIENT~~

In [404] and [403] S. Schwarz developed a semigroup approach to the Fermat and Euler’s theorems. In 1996 M. Lasˇsˇak´ and S, Porubsky´ [240] generalized the idea to ﬁnite commutative rings, and to residually ﬁnite Dedekind domains. (For Dedekind domains, see also W. Narkiewicz [314]). F. Smarandache [430] obtained a generalization of Euler’s theorem by applying a terminating divisibility algorithm. For a best possible variant of this result, with an algebraic extension, see the recent paper by S. Porubsky´ [338]. A generalization of another type of Euler’s theorem, involving Lucas sequences, appears in [349] (see pp. 53-63). This result is based on a Lucas sequence-generalization of Euler’s totient function, and was proved essentially by E. Lucas in 1877 (see [103], p. 398).

~~3 Gauss’ divisibility theorem In 1770 J. L. Lagrange (see [103]) obtained a common generalization of Fermat’s and Wilson’s divisibility theorem:~~

~~p−1 a p−1 − 1 ≡ (a − i)(mod p)(13) i=1 Indeed, for a prime p, and for (a, p) = 1, a p−1 ≡ 0 (mod p). Comparing the constants on both sides of (13) one obtains Wilson’s theorem~~

~~(p − 1)! + 1 ≡ 0 (mod p)(14)~~

For the early history of Wilson’s theorem, and generalizations, see [103]. In a little known paper, N. Gruber [158] has proposed to study (13) for composite n (in place of p − 1) in the following manner:

ϕ(n) a − 1 ≡ (a − a1)(a − a2)...(a − aϕ(n))(mod n)(15) where a1, a2,...,aϕ(n) is a complete system of reduced residues (mod n) (i.e. (ai , n) = 1 for i = 1,ϕ(n).Itisproved in [158] that (15) holds true for all composite n and all a if and only if n = 4orn = 2Fk, where Fk is a Fermat prime (i.e. 2k Fk = 2 + 1, prime). (16) A different kind of generalization of Fermat’s theorem was ﬁrst stated by K. F. Gauss (see [103]) in 1863: n/d Ga(n) = µ(d)a ≡ 0 (mod n) for all a, n.(17) d|n In fact, Gauss proved (17) only for a = prime, while the case a = pm (p prime) 1 is due to T. Schonemann¨ (1846), who showed that G (n) in this case is the number n a

191 CHAPTER 3 of all irreducible polynomials over the finite field of pm elements. The first proof for general a is due to J. A. Serret (1855). In 1880 S. Kantor deduced (17) by proving that 1 G (n) is the number of cyclic groups of order n in any birational transformation of n a order a in the plane. In 1899 L. E. Dickson obtained an inductive proof. He showed also that Ga(n) = ϕ(d)(18) d|(an−1), d(as −1) fors

~~r − ai 1 pi ϕ( ) ai=1 (a n − 1) ≡ 0 (mod n), (22) r = ai where n pi is the prime factorization of n. See also J. Morgado [298]. i=1~~

~~192 THE MANY FACETS OF EULER’S TOTIENT~~

J. Westlund [481] extended Gauss’ theorem (17) to algebraic number fields. Let A be an ideal of an algebraic number field, A = P1 P2 ...Pi , where P1,...,Pi are distinct prime factors of A.LetN(A) be the norm of A.Ifa is an arbitrary algebraic integer, then A| µ(d)a N(A)/N(d) (23) d|A Forfinite permutation groups, S. P. Strunkov [442] has recently proved the following theorem. Let G be a finite permutation group and H asubgroup. We de- fineaMobius-type¨ function on the subgroup of G by recursion: µ(H, H) = 1, and µ(H, T ) =− µ(H, K ). Let T < G and c(T ) denote the number of T -orbits. T >K ≥H Then for any a ∈ Z \{0}, c(T ) µ(H, T )a ≡ 0 (mod |NG (H)|/|H|)(24) T ≥H

4 Minimal, normal, and average order of Carmichael’s function As we have seen, Carmichael’s λ function is closely related to Euler’s totient. For an applet, which computes the values ϕ(n) and λ(n),seethe Internet address: http://www.math-it.de/Mathematik/Zahlentheorie/Zahl/ZahlApplet.html Relations (10) inform us on certain estimates related to ϕ, λ and H.Wenote that the following elementary properties of H are true (see [419])

~~a|b ⇒ H(a) ≤ H(b); (25)~~

max{H(a), H(b)}≤H(a, b) ≤ H(a) + H(b); (26) (a, b) = 1 ⇒ H(a, b) = max{H(a), H(b)} (27) In what follows we shall investigate the minimal, normal, and average order of Carmichael’s function λ. Estimates for the minimal order are implicitly stated in the analysis of the primality testing arguments of L. M. Adleman, C. Pomerance and R. S. Rumely [1]. These are stated explicitly in P. Erdos,¨ C. Pomerance and E. Schmutz [129] as follows: Forany increasing sequence (ni ) of positive integers, and any positive constant c0 < 1/ log 2 one has c0 log log log ni λ(ni )>(log ni ) (28) for sufﬁciently large i. On the other hand, there exists a sequence m1 < m2 < ··· < ... and a constant c1 with c1 log log log mi λ(mi )<(log mi ) for all i.(29)

~~193 CHAPTER 3~~

λ(n) The normal order of magnitude of log was stated without proof in P. Erdos¨ n [109]. A stronger result appears in [129]: There is a set S of positive integers of aymptotic density 1 such that, for all n ∈ S one has −1+ε λ(n) = n/(log n)log log log n+A+O(log log log n) (30) log p where A =−1 + = .2269688 ..., and ε>0isﬁxed, but arbitrarily ( − )2 pprime p 1 small. In [109] appeared also without proof an estimate for the average order of λ:for all ε>0, k > 0andx > x0(ε, k), x 1 x (log log x)k ≤ λ(n) ≤ (31) ( )1−ε log x x n≤x log x

The following sharper result appears in [129]: For all x ≥ 16, we have 1 x B log log x λ(n) = exp (1 + o(1)) (32) x n≤x log x log log log x 1 where B = e−γ 1 − = .34537 ... ( − )2( + ) pprime p 1 p 1 As a corollary of theorem (30) we obtain that ϕ(n) P <(log log n)2, n ∈ S (33) λ(n) where S has asymptotic density 1, and P(r) denotes the greatest prime divisor of r. (For many results on the function P, see [291], Chapter 4). Let N(k) be the number of solutions to λ(n) = k. The following results appear in [129] (without proof):

~~N(k)~~

~~N(k)>exp[exp[(c2 − o(1)) log k/(log log k)]] (35) for inﬁnitely many k (where c2 is a constant).~~

~~194 THE MANY FACETS OF EULER’S TOTIENT~~

~~Similarly, for card{n : λ(n) ≤ x} one has the estimates: (c − o(1)) log x exp exp 2 < card{n : λ(n) ≤ x} < log log x (log 2 + o(1)) log x < exp exp (36) log log x~~

Let Rλ(x) = card{m ≤ x : m = λ(n)}. The exact order of Rλ is not known. Clearly Rλ(x) = o(x)(37) since few numbers have a large divisor of the form p − 1 (see P. Erdos¨ and S.S. Wagstaff, Jr. [132]). On the other hand, x Rλ(x) (38) log x since this is already attained on the primes. The authors of [129] conjecture that

~~c+o(1) Rλ(x) = x/(log x) (39) where c is a constant. Probably c ∈ (0, 1),but the correct value of c remains in doubt.~~

5Divisibility properties of iteration of ϕ Certain properties of the iteration of ϕ were included in [291], Chapter 1. Here we will study the divisibility properties of ϕ( j), where ϕ(1)(n) = ϕ(n), ϕ( j)(n) = ϕ(ϕ( j−1)(n)) ( j > 1),aswell as of certain related functions. W. Sierpinski [414] determined all n such that

~~ϕ(n)|n (40)~~

It is not difﬁcult to see that all solutions of (40) are n = 1, and n = 2a · 3b, where a > 0, b ≥ 0. (41) F. Smarandache [431] has considered mϕ(n)|n, for a ﬁxed m,but clearly this implies (40). In 1982 M. Hausman [193] completely described the solutions of

~~ϕ( j)(n)|n (42)~~

~~This was rediscovered in 1989 by D. Marcu [284]. However, in 1988 V. Siva Rama Prasad and Ph. Fonseca [423] have studied a general class of functions, including (42).~~

~~195 CHAPTER 3~~

~~For j = 2, the solutions of (42) are~~

~~n = 1, 2a, 3, 2a · 3b, 2a · 5, 2a · 7, where a, b ≥ 1 (43)~~

~~This has been rediscovered by L. Toth´ [454], too. A problem proposed by A. H. Stein [438] asks for the determination of all pairs of positive integers m, n such that~~

~~ϕ(m)|n and ϕ(n)|m (44)~~

For m = n this generalizes Sierpinski’s problem (40). It is not difﬁcult to see that the relations of (44) imply that ϕ(ϕ(n))|n, ϕ(ϕ(m))|m so n, m, must be in the sets of (43). Another method is based on the fact that all solutions can be generated by primitive solutions (m, n), i.e. when gcd(m, n) = squarefree. Since ϕ(pn) = pϕ(n) for p|n, p prime, all other solutions can be obtained from primitive ones by the following rule: If p is a prime dividing both m and n, then the pair (m, n) is a solution iff the pair (pm, pn) is a solution. There are eleven primitive pairs of solutions, namely: (1,1), (1,2), (2,2), (2,3), (2,4), (2,6), (4,6), (4,10), (6,6), (6,14) and (6,18). (45) In 1950 H. N. Shapiro [411] introduced a class F of arithmetic functions f which r ( ) = < = ai are of the form: f 1 1andif 1 n pi is the prime factorization of n, then i=1

r ai −1 f (n) = f (pi )[A(pi )] (46) i=1 where f (pi ), A(pi ) are integers satisfying 0 < f (pi )1for odd primes pi (i = 1, r), while f (2) = 1, A(2) = 2. Since f (n) 2, by denoting by f (k) the kth iterate of f ,for n > 2 there is a unique integer k = k f (n) such that

~~f (k)(n) = 2.(47)~~

~~Deﬁne k f (1) = k f (2) = 0, and put k f (n), if n is odd, c f (n) = (48) k f (n) + 1, if n is even~~

~~Shapiro proved that for any f ∈ F, the function c f is completely additive, i.e. c f (mn) = c f (m) + c f (n) for all m, n;iff f (p) is even, and c f (2A(p)) = c f (p) + 1 for all odd primes p.(49)~~

~~196 THE MANY FACETS OF EULER’S TOTIENT~~

~~Let now f ∈ F such that (49) is true, and let {qi } denote the sequence of odd primes in increasing order. Let F1 be the class of all f ∈ F such that~~

f (qi ) = qi − εi and A(qi ) = qi (50) where (εi ) are odd integers satisfying 0 <εi < qi , i = 1, 2, 3,... Remark. If εi = 1 for all i, then by (46) and (50) one gets f = ϕ,sothat ϕ ∈ F1. Clearly, the set F1 is uncountable. Let f ∈ F1,anddeﬁne

( j) S f ( j) ={n ∈ N : f (n)|n}, i.e. the set of n such that (42) holds true. (51) Let Pf ( j) denote the set of primitive solutions of (42), i.e. n P ( j) = n ∈ N : n ∈ S ( j) and ∈ S ( j) (52) f f 2 f Siva Rama Prasad and Fonseca [423] have proved the following result: For f ∈ F1, j ≥ 1 one has ∞ i S f ( j) = 2 Pf ( j); (53) i=0 therefore it is enough to characterize the primitive solutions. For these, the following theorem is valid: Let n ∈ Pf ( j).Then:

~~c f (n)< j iff n is odd; (54)~~

~~c f (n) = j iff n = 2m, where m is odd, and c f (m) = j − 1; (55) l c f (n)> j iff n = 2(εi + 2) , for some i and l ≥ j.(56)~~

(Here c f (n) is given by (48)). Since ϕ ∈ F1, results (53)-(56) are extensions of the theorems by Sierpinski, Hausman. Another generalization of (42) has been studied by F. Halter-Koch and W. Steindl [175], namely ϕ( j)(n)|Qn (57) where Q is a ﬁxed positive integer. Let

~~( j) SQ( j) ={n ∈ N : ϕ (n)|Qn},(58) n P ( j) = n ∈ N : n ∈ S ( j), ∈ S ( j) (59) Q Q 2 Q~~

~~197 CHAPTER 3 and~~

~~C(n) = min{ j ≥ 0: ϕ( j)(n) ≤ 2} (60)~~

~~Since~~

k SQ( j) ={2 n : m ∈ PQ( j), k ≥ 0},(61) we need a characterization of primitive solutions given by (59). It can be shown that if C(n)< j, then n ∈ SQ( j) and n ∈ PQ( j) only when 2 n.IfC(n) ≥ j, Q odd, and n ∈ PQ( j), then n ≡ 2 (mod 4).Forodd n one has n ∈ S2Q( j) \ SQ( j) only if 2n ∈ PQ( j).(62) If Q is odd, then if n ∈ PQ( j) and C(n) ≥ j, then n ≡ 2 (mod 4).Ifn ≡ 2 (mod 4),andC(n) = j, then n ∈ PQ( j).Sowemay assume in what follows C(n)> j.Forodd Q the following complete characterization is true (see [175]): Let Q be odd, and suppose C(n)> j.Then n ∈ PQ( j) if and only if one of the following cases holds true.

a) n = 2 · 3s with s ≥ j + 1; b) n = 2p, where p is a prime of the form p = 2 · 3s + 1, with s ≥ max{2, j} and 3s+1− j |Q; c) n = 2 j (p + 1) − 2, where p > 3isaprime such that p − 1 |(2 j − 1)Q, and the numbers 2r p + 2r − 1 (r = 0, 1,..., j − 1) 2 are all primes. (63) When Q is even,asimilar characterization is an open question. By assuming Schinzel’s Hypothesis H (see e.g. [414]) for linear polynomials, the following can be proved (see [175]): Let p be a prime, and put L = C(p).Let j ≥ L + 2. Then there exist inﬁnitely L many even numbers Q such that 2 Q and for each such Q there is an n ∈ PQ( j) with p|n,butp Q.(64) For certain remarks on the divisibility properties of the iteration of Euler’s totient, see also W. Steindl [439]. By studying certain asymptotic properties of functions like ϕ( j)(n)/ϕ( j+1)(n),P. Erdos,¨ A. Granville, C. Pomerance and C. Spiro [124] (see also [291] for details) proved also certain results on prime factors of ϕ( j)(n).For example: There is a positive absolute constant c > 0, such that the set of natural numbers n,for which there is some j with ϕ( j) divisible by every prime up to (log n)c,has asymptotic density 1. (65)

~~198 THE MANY FACETS OF EULER’S TOTIENT~~

∞ Let (n) = n ϕ( j)(n). Then j=1 log n ω( (n)) ≤ (66) log 2 (Here ω(m) denotes the number of distinct prime factors of m). In particular, for all n there is some prime p, with p log n log log n such that

~~p (n)(67)~~

The following two conjectures are also stated in [124]: 1) For each prime p, let N(x, p) denote the number of n ≤ x with p| (n). Then for every ε>0, N(x, p) = o(x) uniformly for p >(log x)1+ε;(68) and N(x, p) ∼ x uniformly for p <(log x)1−ε.(69) 2) For each ε>0, the upper asymptotic density of the set {n ∈ N : ω(ϕ( j)(n)) > nε} tends to 0 as j →∞ (70) The authors state (without proof) that by sieve methods they can prove (70) for ε>2/3. Results related to the distribution of the distinct prime divisors of ϕ(n) are summarized in [291]. Recently these have been extended to the iteration function ϕ( j)(n) as follows (see N. L. Bassily, I. Katai´ and M. Wijsmuller [29]): √ Foreach fixed integer j ≥ 1 let a j = 1/( j + 1)!andb j = 1/ 2 j + 1 · ( j!). Then for each real number z, 1 ω(ϕ( j)(n)) − a (log log x) j+1 lim card n ≤ x : j < z = →∞ j+1/2 x x b j (log log x) z 1 2 = √ e−t /2dt (71) 2π −∞ For k = 1, (71) is proved in P. Erdos¨ and C. Pomerance [127] and M. Ram Murthy and V. Kumar Murthy [346]. The case k = 2itwasprovedbyI.Katai´ [219] (see also [291]). For the distribution of ϕ( j)(p − 1), where p is a prime, the following is true (see [29]): Foreach fixed integer j,anda j , b j defined as in (71), one has 1 ω(ϕ( j)(p − 1)) − a (log log x) j+1 lim card p ≤ x : j < z = →∞ j+1/2 x π(x) b j (log log x) z 1 2 = √ e−t /2dt (72) 2π −∞

~~199 CHAPTER 3~~

For k = 1 this has been previously proved by I. Katai´ [220]. Let now λ( j) be the jth iterate of the Carmichael function and for n ≥ 1 let g(n) be the least positive integer j such that λ( j)(n) = 1. (73) By using some divisibility properties of the λ function, e.g.

~~λ(ab) = [λ(a), λ(b)] for (a, b) = 1 (74)~~

(implying a|b ⇒ λ(a)|λ(b), which is similar to the corresponding property of Euler’s totient ϕ), I. Niven [322] proved the following fundamental properties of the function g: (the ﬁrst is similar to (27))

~~g(ab) = max{g(a), g(b)} for (a, b) = 1; (75)~~

g(22n) = g(22n+1) = n + 1, g(pn) = n − 1 + g(p), for any odd prime p, n ≥ 1.(76) Now let C(n) be the integer j such that ϕ( j)(n) = 2 (n ≥ 3), with C(1) = C(2) = 0, be the Shapiro-Pillai function (see [291], p. 34). Then (see [322])

lim sup(g(n + 1) − g(n)) = lim sup(C(n) − g(n)) =+∞ (77) n→∞ n→∞ and lim inf(g(n + 1) − g(n)) =−∞, lim inf(C(n) − g(n)) =−1 (78) n→∞ n→∞ We note that, in a little known paper, A. Muranyi´ [305] introduced a function in an essentially different manner as Shapiro, Pillai or Niven. For a positive integer n > 1letm = m(n) be the integer such that ϕ(m)(n) = 2ifn is odd, and ϕ(m)(n) = 1 if n is even. Put m(1) = 0. Then (unlike the function C(n)) m(n) is completely additive function (see [305]), i.e.

~~m(ab) = m(a) + m(b) for all a, b ≥ 1 (79)~~

~~Let Mk ={n ∈ N : m(n) = k}, k ﬁxed. If a ∈ Ms and b ∈ Mk, with s ≥ k and b|a (i.e. b divides a), then writing a = b ·t one has (for b = ﬁxed)~~

min t = 2s−k, max t = 3s−k (80) which give the minimum, respective maximum values of a ∈ Ms divisible by a ﬁxed b ∈ Mk. Another result states that the number of elements of Ms which are divisible by a given integer a,isequal to the number of elements of the set Ms−m(a). (81)

~~200 THE MANY FACETS OF EULER’S TOTIENT~~

~~For the function m the following estimates are true:~~

~~log n log n ≤ m(n) ≤ (82) log 3 log 2~~

(82) implies ax log x ≤ m(n) ≤ bx log x (82) n≤x with a, b > 0 constants. It is not known if m(n) ∼ cx log x (82) n≤x for certain constant c (see I. Z. Ruzsa [359]). Finally we note that Muranyi´ used the notation m(n) = grav(n) (i.e. the ”gravaritm” of n), in concordance with the logarithm of n.

~~6 Congruence properties of ϕ and related functions We now study other congruence properties of Euler’s totient and related functions. Perhaps, the simplest congruence satisﬁed by ϕ(n) is~~

~~ϕ(n) ≡ 0 (mod 2) for n ≥ 3 (83)~~

From the known form of ϕ (see formula (4) of 3.1) it follows that if for each prime p|n ⇒ p|m, then ϕ(mn) = nϕ(m)(84) Particularly, if n = p (prime), and p|m, then ϕ(mp) = pϕ(m),aformula frequently used in the theory of Euler’s totient. This is due to J. A. Grunert (see [103], p. 117). Another simple, but useful divisibility property of ϕ is

~~m|n ⇒ ϕ(m)|ϕ(n)(so,ϕ(m) ≤ ϕ(n)) (85)~~

~~As a consequence of (85),~~

ϕ(m)ϕ(n)|ϕ(mn) ⇔ gcd(m, n) ∈{1}∪{2a · 3b : a > 0, b ≥ 0} (86) d Indeed, by the identity ϕ(mn) = ϕ(m)ϕ(n) , where d = gcd(m, n) (due to ϕ(d) E. Prouhet ([103], p. 118), we must have ϕ(d)|d, and this is Sierpinski’s relation (40),

~~201 CHAPTER 3 having solutions given by (41). A similar property is ϕ(mn) = (m, n)ϕ([m, n]), due to E. Lucas [272]. A counterpart of (85) is~~

~~m|n, m < n ⇒ n − ϕ(n)>m − ϕ(m)(87)~~

~~(see I. Niven and H. S. Zuckerman [323] (problem 22 of 2.4)). In 1913 U. Scarpis (see [103]) proved that if a is a prime, then~~

~~n|ϕ(an − 1) for all n ≥ 1, a ≥ 2.(88)~~

In fact, (88) is true for all positive integers a. Indeed, since an ≡ 1 (mod an − 1), (a, an − 1) = 1, but as ≡ 1 (mod an − 1) < ∗ for s n,sothe order of a in Zan−1 (the multiplicative group of reduced classes (mod an − 1))isn, which divides the order of the group, i.e. ϕ(an − 1). More generally, N. G. Guderson [164] proved that if a > b and m = γ(n) = greatest squarefree divisor of n, then n2 |ϕ(an − bn), and n|ϕ(an + bn)(89) γ(n) See also P. Kesava Menon [221] for related results. See also a problem by R. E. Shafer [409] for the second relation of (89). In 1961 A. Rotkiewicz [356], by using (85) and the Zsigmondy-Birkhoff-Vandiver theorem on prime divisors of an −bn (see e.g. [349], p. 43) deduced that

T (n)|ϕ(an − bn) if a > b, n ≥ 1 (90) where T (n) denotes the product of divisors of n.(Infact, T (n) = nd(n)/2, where d(n) is the number of divisors of n,see Chapter 1). M. Deaconescu and J. Sandor´ [97] (see also J. Sandor´ ([363], pp. 232-233)) have considered functions f : N → N such that

~~n| f (an − 1) for all n, a > 1.(91)~~

Relation (91) is true iff ϕ(n)| f (n) for all n. Related to the factors of an − bn is also the following non-divisibility property (see [364], ([363], pp. 230-231)). Let a > b, (a, b) = 1. Suppose that (a, n) = (b, n) = (a − b, n) = 1. Then

~~n (an − bn), and n (an−ϕ(n) − bn−ϕ(n))(92)~~

~~Particularly, n (2n−ϕ(n) − 1) for any n > 1.(93)~~

~~202 THE MANY FACETS OF EULER’S TOTIENT~~

~~An improvement of (88) is due to H. Gupta [169]: For all a ≥ 2, p prime and j ≥ 1 one has~~

~~j p j( j+1)/2|ϕ(a p − 1)(94)~~

This implies (88). Indeed, writing n = kpj with (k, p) = 1, since p j |ϕ(b p j − 1) for all b ≥ 2, putting b = ak one gets p j |ϕ(an − 1),and the result follows. Another improvement of (88) is the following divisibility property (see M. Dea- conescu and J. Sandor´ [98]):

~~ n|ϕ(φn(a)) for all n ≥ 1, a ≥ 2 (94 )~~

n where φn denotes the nth cyclotomic polynomial. Since φn(a)|(a − 1),by(85) we get that (88) is a consequence of (94) (by the transitivity property of ”|”). Another reﬁnement is an − 1 m|n, m < n ⇒ n|ϕ |ϕ(an − 1), (94) am − 1 see [98]. There are also some congruence properties of the totient function, which are connected to other number-theoretical functions. Forexample, M. V. Subbarao [443] considered the relations

~~ϕ(n)|(nσ(n) − 2), (95)~~

n|(ϕ(n)d(n) + 2)(96) Clearly, (95) and (96) are satisfied by any n = p (prime). On the other hand, it is not difficult to see that all composite solutions of (95) are n = 4, 6 and 22. (97) The determination of all composite solutions of (96), however, is an Open problem (see [443]). Though n = 4isasolution, no other solution is known. If n > 4 composite satisfies (96), Subbarao proves that n must have at least 4 distinct prime factors; n is squarefree; if p is an odd prime divisor of n, then there is no prime divisor of the form px + 1; if ϕ(n)d(n) + 2 = Kn, then K and n are of opposite parity, and 4 K .(98) Also, for 1 ≤ K ≤ 1023 the possible number of prime divisors of n is studied. M. Zhang [486] considers the divisibility

n|(ϕ(n) + σ(n)), (99) and proves that numbers of the form n = pα (α > 1) or n = pαq (α ≥ 1), where p, q are distinct primes, are not solutions of (99). For n < 107 there are 17 composite solutions. (Clearly, all n = p are solutions). The following Open problems are stated:

~~203 CHAPTER 3~~

a) Is there a solution n of (99) with ω(n) = 2? b) Is there an odd composite solution? c) Is there a solution n ≡ 2 (mod 6)? d) Are there inﬁnitely many composite solutions? (100) For the values of ϕ at Fibonacci, or Lucas numbers, S. Singh [418] proves the following (note that 4|ϕ(Fn) for all n ≥ 5, see C. Kimberling [224] (see also V. E. Hogatt and H. Edgar [113]))

~~2n|ϕ(F2n), if n = 1, 2, 3, 4, 8, 16; (101)~~

~~80n|ϕ(F5n), if n > 1isodd; (102)~~

~~n|ϕ(Ln), if n > 3isodd; (103)~~

~~10n|ϕ(L5n), for all n; (104) n|ϕ( ), .() 2 L2n+1 for all n 105~~

~~The congruences like ϕ(F2n+1) ≡ 0 (mod 2n + 1)), ϕ(L2n) ≡ 0 (mod 2n) are not true in general. It is conjectured that~~

~~n 2 |ϕ(L2n ) for inﬁnitely n (106)~~

7 Euler’s totient in residue classes K. Ford [137] remarks that the following congruences are true: ϕ(n) If n > 6, then ≡ 1 (mod 6) iff n = p2α or 2p2α,(107) 2 where α is a positive integer, and p is a prime, p ≡ 11 (mod 12). Similarly, ϕ(n) ≡ 3 (mod 6) is n = pα or 2pα,(108) 2 where α is a positive integer, and p ≡ 7 (mod 12) is a prime; or else p = 3 and α>1. ϕ(n) ≡ 5 (mod 6) iff n = p2α−1 or 2p2α−1,(109) 2 where α is a positive integer, and p ≡ 11 (mod 12) is a prime. ϕ(n) Relations (107)-(109) together cover all residues of (mod 6). Similar con- 2 gruences can be determined for any modulus of the form 2q, where q is an odd prime, or more generally to ﬁnd all integers n such that

~~ϕ(n) ≡ k (mod 2q), (110) 2~~

~~204 THE MANY FACETS OF EULER’S TOTIENT where k is odd (see [137]). Let Sm,r denote the set of integers n with~~

~~ϕ(n) ≡ r (mod m)(111)~~

2α Then e.g. (107) can be rewritten as S12,2 ={3, 4, 6}∪{n : n or n/2 = p } where p ≡ 11 (mod 12), etc. Let N(x, m, r) = card{n ∈ Sm,r : n ≤ x},(112) which counts how many members Sm,r has up to x. P. Erdos¨ (see [102]) proved the asymptotic result:

N(x, m, 0) ∼ x as x →∞,(113) for any integer m. The following asymptotic results are considered in the cases m = 12 or m = 3 (see T. Dence and C. Pomerance [102]) √ 1 1 x N(x, 12, 2) ∼ + √ ; (114) 2 2 2 log x x N(x, 12, 4) ∼ c1 ; (115) log x 3 x N(x, 12, 6) ∼ · ; (116) 8 log x x N(x, 12, 8) ∼ c2 ; (117) log x 3 x N(x, 12, 10) ∼ · ; (118) 8 log x x N(x, 3, 1) ∼ c1 ; (119) log x x N(x, 3, 2) ∼ c2 (120) log x where √ 2 3 − / 1 c = c 1 2(2c + c ) and c = 1 + , 1 π 3 3 4 3 2 − 3 p≡2(3) prime p 1 √ 1 2 3 − / c = 1 − , c = c 1 2(2c − c )(121) 4 ( + )2 2 π 3 3 4 p≡2(3) prime p 1 3

~~205 CHAPTER 3~~

Dence and Pomerance prove also (inspired by an argument by W. Narkiewicz [315]) that if the residue class r (mod m) contains a multiple of 4, then it contains infinitely many numbers ϕ(n).(122) In what follows, by a ”totient” we mean a value taken by Euler’s function ϕ(n). Since 1 is the only odd totient, by result (122) of Dence and Pomerance, it remains to examine residue classes consisting entirely of numbers ≡ 2 (mod 4).K.Ford, S. Konyagin and C. Pomerance [143] have characterized which of these residue classes contain infinitely many totients and which do not. The following negative result is true: For any ε>0 there exist such m that at least (1 − ε)m residue classes a (mod 4m),0< a < 4m, a ≡ 2 (mod 4) are totient-free. (123) As a corollary of (123) one gets that the union of all totient-free residue classes has density 3/4. (124) On the other hand, holds true a positive result: The set of all odd numbers m such that for any s ≥ 1andforanyevena the residue class a (mod 2sm) contains infinitely many totients, has a positive lower density. (125) If a residue class r (mod m) contains infinitely many totients, the authors in [143] remark that using the methods of [102] and [316], it is possible to get asymptotic formulas for N(x, m, r).Aresult of H. Delange [101] can be used to give an asymptotic formula for the number of n ≤ x for which m ϕ(n) (where m is fixed). For the distribution in residue classes of values of general multiplicative functions we quote also [315], [316], [101]. It is well known that the set of values of the function ϕ (i.e. the set of totients) has (asymptotic) density 0 (see [323], section 11.3). H. Berekova´ and T. Salˇ at´ [38] prove that the sets of positive values of the functions ϕ + d and ϕ − d have density 0. R. B. Killgrove [223] poses the problem if the set of values of ϕ is or not a Dirichlet set?

~~8 Prime totatives If we consider the ϕ(30) = 8 integers that are ≤ 30 and relatively prime to 30, we ﬁnd these to be 1, 7, 11, 13, 17, 19, 23, 29.~~

Thus, apart from 1, all of these integers are primes. It is interesting that 30 is the largest integer with this property, i.e.: The number 30 is the largest integer such that all the integers less than it that are relatively prime to it are prime (apart from 1). (126) Usually, this result is attributed to H. Bonse [46] (who obtained a proof based on ... ≥ 2 his inequality p1 p2 pk pk+1, where pk is the kth prime (see [291], Chapter 7)).

~~206 THE MANY FACETS OF EULER’S TOTIENT~~

However, this was ﬁrst proved by S. Schatunowsky (see [103], p. 132). E. Maillet [278] gave a generalization, as a solution to an Open problem posed by G. de Roc- quigny [355] (see also [103]). In 1984 H. Iwata [207] rediscovered this result which can be stated as follows: There are only ﬁnitely many integers b such that all smaller positive integers n, with (n, b) = 1, are products of at most k prime factors. (127) In 1989 L. Cseh [91] shows that if k = 1 and n, b are restricted to be odd integers, then the largest b is 105. (128) This was an Open question of S. W. Golomb. In fact, Cseh extends the Maillet (Iwata) theorem for Beurling’s generalized primes (see Chapter 2). We note that Bonse [46] proved Maillet’s theorem (127) for k = 1, 2, 3 without using Chebysheff’s theorem, which states that there is always a prime between a and 2a (a ≥ 2). Later, in 1909, R. Remak (see [103], p. 138) proved completely (127) for all k, without using the Chebysheff theorem. Another proof was obtained by E. Landau (see [103], p. 138). For integers a such that (a, n) = (a + 1, n) = 1. L. Carlitz [61] showed the following: Let n be odd. Then ≡ 1 (mod n) and ≡−1 (mod n)(129) 1≤a

For k ≥ 1aﬁxed integer, let Sk(n) denote the number sets of k consecutive integers each less than n and relatively prime to n (this is a generalization of Euler’s totient). For 1 ≤ r ≤ k let P denote the product of the rth terms of the Sk(n) sets of consecutive integers described above. V. Schemmel [391] proved that − − ( ) pk 1 ≡{(−1)r 1(r − 1)!(k − r)!}Sk n (mod n)(131) For r = 1, k = 2 this becomes P ≡ 1 (mod n). Forsquarefree integers which are relatively prime to n, the following result has been proposed by E. B. Cossi [89]. Let R(n) denote the number of ordered pairs of squarefree integers j, k such that j + k = n and ( j, n) = (k, n) = 1. Then

R(n)>cϕ(n) for n ≥ n0.(132) In fact, J. Herzog and P. R. Smith (see the solution from [89]) showed (132) with 1 c = , n = 6000000. Since R(n) ≥ 2 for 2 < n ≤ 6000000 (see above), (132) 200 0 holds with n0 = 1 for a suitable constant c > 0.

~~207 CHAPTER 3~~

~~9 The dual of ϕ, noncototients Relation (11) gives Redei’s´ generalization of Euler’s theorem. This relation involves the dual function~~

~~ϕ(n) = n − ϕ(n), n ≥ 1 (133)~~

~~Certain divisibility properties of ϕ,similar to that of ϕ, are introduced in L. Panaitopol [327]. For example,~~

~~ϕ(n)|n iff n = pα (p prime,α≥ 1). (134)~~

The relation ϕ(m) ϕ(n) = is true iff γ(m) = γ(n), (135) m n where γ(m) denotes the core of n (i.e. greatest squarefree divisor of n; see [291], Chapter 6). Let l(n) = (ϕ(n), ϕ(n + 1)), n ≥ 3.(136) Then for any odd number m, there exist inﬁnitely many integers n such that

~~m|l(n)(137)~~

~~The following Open question is stated: If k is a positive integer, and m is odd, is there an n such that~~

~~m|(ϕ(n), ϕ(n + 1), . ..,ϕ(n + k))? (138)~~

If n is prime, clearly ϕ(n)|(n − 1)(139) On the other hand, there are many composite solutions of (139). For example, n = 15, 85, 225 259, 391, etc. In fact, there are 21 such composite numbers n with n < 105.Itisproved that all n satisfying (139) must be odd and squarefree; n − 1 h = must be even, and P(n)>hω(n) (140) ϕ(n) where P(n) is the greatest prime factor of n. In 1959 W. Sierpinski asked, whether there are inﬁnitely many integers not of the form ϕ(n) (i.e. noncototients, see [171], p. 91). Recently, J. Browkin and A. Schinzel [53] have shown that the equation

~~ϕ(x) = 2k · 509203 (k = 1, 2,...)~~

208 THE MANY FACETS OF EULER’S TOTIENT has no solutions. The following Open problem is stated: do the integers not of the form ϕ(n),haveapositive lower density? It is not known if there exist inﬁnitely many composite solutions of (139). (141) A number of asymptotic results involving ϕ are included in [328]. For a ﬁxed positive integer m, ϕ(n) m (k − 1)! x2 = x2 + O .(142) ϕ( ) k( )k m+1 2≤n≤x n k=1 2 log x log x

~~For any a > 0, 1 m (k − 1)! x = x + O .(143) (ϕ( ))a k m+1 n≤x n k=1 log x log x~~

Forafixed positive integer k one has 1 4 ϕ(n) · ϕ(n + k) = x2 c(k) + − + O(x2 log2 x), (144) π 2 n≤x 3 1 2 1 where c(k) = 1 − 1 + . 2 ( 2 − ) 3 pprime p p|k p p 2 In 1980 P. Erdos¨ [110] (see also R. K. Guy [171]) has conjectured that: i)ϕ(n)>ϕ(ϕ(n)) for almost all n; (145) ii)ϕ(n)<ϕ(ϕ(n)) for infinitely many n. Recently, A. Grytczuk, F. Luca and M. Wojtowicz´ [160] have settled ii), by proving that there exists an infinite sequence of positive integers (nk) such that

~~k ϕ(nk) + 2 ≤ ϕ(ϕ(nk)) (146)~~

For conjecture i) of (145) it is proved that if n > 2issuch that the odd part of n is squarefull, then n satisﬁes the inequality of i). (147) Particularly, the lower asymptotic density of numbers n with inequality of i) is ≥ 0.54. (148) It is immediate that ϕ(n) − ϕ(ϕ(n)) lim sup = 1 (149) n→∞ n

~~The exact value of lim inf,however is an Open question (see [160]). n→∞~~

~~209 CHAPTER 3~~

~~10 Euler minimum function The Euler minimum function is deﬁned by~~

~~E(n) = min{k ≥ 1: n|ϕ(k)} (150)~~

It was introduced by P. Moree and H. Roskam [297], and independently by J. Sandor´ [365] (see also [363], pp. 141-149) as a particular case of the more general function A( ) = { ∈ | ( )},(⊂ N)() F f n min k A : n f k A 151 if such a function exists. For A = N, f = ϕ one obtains the function E given by (150). Since by Dirichlet’s theorem on arithmetical progressions there exists a ≥ 1 such that k = an + 1isprime, by ϕ(k) = an is a multiple of n,soE(n) exists. We note that for A = N, f (k) = k! one obtain the Smarandache function, while for A = P (set of primes), f (k) = k! one gets a new function [365]. There is also a ”dual”of(151), namely

A( ) = { ∈ ( )| },() Gg n max k A : g k n 152 if this is well deﬁned. For A = N, g(k) = k! this is denoted by S∗(n) and studied in [365]. For f (k) = ϕ(k), A ⊂ N in (151) one obtains a generalization of the Euler minimum function, while from (152) one can introduce the Euler maximal function by (see [366]) E∗(n) = max{k ≥ 1: ϕ(k)|n} (153) Since ϕ(1) = 1|n for all n, this function is well deﬁned. It can be proved that (see [366]) that

max{E(pa), E(qb)}≤E(pa · qb) ≤ [E(pa), E(qb)] for p = primes (154) so one needs to study ﬁrst E on prime powers. Let q be a prime, and denote by A(q) the set of squarefree numbers composed of only primes p satisfying p ≡ 1 (mod q).For convenience, let the empty set have minimum equal to ∞. Then Moree and Roskam prove ([297]):

E(qn) = min{m, qn+1}, where m = min{a ∈ A(q) : qn divides ϕ(a)/qn < q} (155) For (u,v) = 1 let p(v, u) denote the smallest prime with p ≡ u (mod v) and pi (v, u)(i ≥ 2) the ith smallest such prime. As a corollary of theorem (155) one has: The largest prime divisor of ϕ(E(qn)) is q; The smallest prime divisor of E(qn) is ≥ q;

~~210 THE MANY FACETS OF EULER’S TOTIENT~~

~~log q If q is odd, then ω(E(qn)) < min n + 1, ; (156) log 2~~

~~E(q) = min{q2, p(q, 1)};~~

2 3 2 E(q ) = min{q , p(q , 1), p(q, 1)p2(q, 1)}. This corollary shows that the behaviour of E is strongly related to the distribution of primes in arithmetic progressions. Another consequence of (155) is that

~~E(pa) = E(qb) if p = q (primes); (157) E(pa) = E(pb) if a = b~~

The prime 2 has the property that E(2n) = 2n+1 for infinitely many n. Let M denote the set of primes p such that E(pn) = pn+1 for infinitely many n. Let q be an odd prime, and suppose that there are integers a, d, n0 such that n n+1 n E(q ) = min{q , p(q , 1)} for n ≥ n0,andn ≡ a (mod d). Then q ∈ M. (158) It can be proved that 2, 3, 5, 7, 13, 19, 31 ∈ M, and in [297] it is conjectured that every prime is in M. Another function, connected also to the Euler minimum function is the Euler l.c.m. sequence,definedby

~~ek = lcm(ϕ(1), ϕ(2),...,ϕ(k)), k = 1, 2,... (159)~~

ek Let ak = (k ≥ 2).Wesaythat k ≥ 2isajumping point if ak > 1. Then ek−1 the following interesting fact is true: The number k ≥ 2isajumping point iff k = E(pr ) for some prime p and positive integer r > 1. (160) This implies that for k ≥ 2, ak is a prime, or equals 1. (161) For the growth of the Euler l.c.m. sequence we have (see [297]):

.6687 exp(k ) ek exp((1 + ε)k)(162) for ε>0ﬁxed, but arbitrary. We note that the result on the functions E and e are used in [297] in the study of the discriminator D( j, n) = smallest positive integer k for which the ﬁrst njth powers are distinct modulo k.This notion has been introduced by L. K. Arnold, S. J. Benkoski and B. J. McCabe [15]. (For more References, see [297]). The study of the discriminator involves also the Fermat’s quotient of a prime p, related to the congruence a p−1 − 1 ≡ 0 (mod p2).

~~211 CHAPTER 3~~

For many results, as well as open problems on Fermat’s quotients, see e.g. [349] (pp. 335-346). Recently, the Fermat’s quotients have been extended to composite moduli m by A. Takashi, K. Dilcher and L. Skula [452] who considered the congruence aϕ(m) − 1 ≡ 0 (mod m2)(163) for composite m.Ifm is such an integer, then P(m) = largest prime factor of m, satisﬁes the congruence

~~a P(m)−1 − 1 ≡ 0 (mod (P(m))2)(164)~~

~~(i.e., is a Wieferich prime). With a given P(m), all solutions of (163) can be determined (see [452]).~~

11 Lehmer’s conjecture, generalizations and extensions Finally, we shall deal with a famous divisibility problem on Euler’s totient, namely Lehmer’s conjecture.Wewill include also some analogues problems on Euler’s or related totients. In 1932 D. H. Lehmer [250] considered the relation

ϕ(n)|(n − 1), (165) and asked if there are any composite solutions. This problem is still open (see e.g. [171]), ”many people feeling it is as difﬁcult as the odd perfect number problem” (quotation from C. Pomerance [333]). Let n be a composite solution of (165). Lehmer proved that then n must be squarefree, odd, and ω(n) ≥ 7 (where ω(n) denotes the number of distinct prime factors of n). Further, he showed that if pq|n (p, q distinct primes), then q ≡ 1 (mod p).(166) In 1944 F. Schuh [399] proved that if 3|n, then

k ≡ 1 (mod 3), (167) where n − 1 k = .(168) ϕ(n) Clearly, k = 1iffn = prime. He ”proved” also that ω(n) ≥ 11, (169) but this proof was not correct. In 1970 E. Lieuwens [256] corrected Schuh’s proof, so result (169) was established. Lieuwens proved also that if 3|n, then

~~ω(n) ≥ 212 and n > 5.5 · 10570 (170)~~

~~212 THE MANY FACETS OF EULER’S TOTIENT~~

Similarly, he proved that if 3 n,5 n (i.e. the least prime factor of n is ≥ 7), then ω(n) ≥ 13. (171) In 1977 M. Kishore [225] proved without any condition that (171) is true, i.e. ω(n) ≥ 13. (172) Even Lehmer ([250]) showed that if k = 3in(168), then ω(n) ≥ 32, (173) and Kishore proved that for any k > 2 one has ω(n) ≥ 33. (174) In 1980 G. L. Cohen and P. Hagis Jr. [86] improved result (171) to ω(n) ≥ 17, and result (172) to ω(n) ≥ 14. (175) D. W. Wall [472] improved (171) to ω(n) ≥ 26. (176) Cohen and Hagis have obtained without any condition the bound

n > 1020,(177) which apparently is the best current result of this type. In 1985 M. V. Subbarao and V. Siva Rama Prasad [445] have studied the analogue of Lehmer’s problem to the unitary totient function ϕ∗, i.e.

ϕ∗(n)|(n − 1)(178) Since ϕ∗(n) = (pα − 1),itisimmediate that ϕ∗(n) = n − 1ifandonly if n is pα n aprime power: n = pα (p prime, α ≥ 1). We shall call n a proper solution of (178), if ϕ∗(n) = n − 1, i.e. when n − 1 k∗ = > 1 (179) (ϕ∗(n) Let S denote the set of composite solutions of (165) and S∗, the set of proper solutions of (178). Since S contains only squarefree numbers n, and ϕ(n) = ϕ∗(n), for such n, clearly S S∗ (180) i.e. S is a proper subset of S∗. Subbarao and Siva Rama Prasad prove that any solution is odd, and not a powerful number; (181) if 3|n, then ω(n) ≥ 1850. (182) By (180), relation (182) improves on Lieuwen’s bound on ω(n) given by (170). Further, if 3 n, 5 n,then ω(n) ≥ 11; (183) if 3 n, 5 n, then ω(n) ≥ 17; (184) if k∗ = 3, 4, 5in(179), then ω(n) ≥ 33. (185) Similarly: If k∗ = 5andn is squarefree, then ω(n) ≥ 53. (186) If k∗ = 6, then ω(n) ≥ 140 or 48, according as n is squarefree, or not. (187)

~~213 CHAPTER 3~~

If k∗ = 7, then ω(n)>200 or 103 according as n is squarefree, or not. (188) If k∗ ≥ 8, then ω(n)>200. (189) Another result says that if 2 <ω(n) ≤ 16, then k∗ = 2, 3 n,5|n and 7|n. (190) Let Jk be the Jordan totient function (see (42) of 3.1). The analogue of Lehmer’s problem for Jk can be easily settled. Namely, it is not difﬁcult to prove (see [445]) that if k > 1, then k Jk(n)|(n − 1) iff n = prime (191) In 1989 V. Siva Rama Prasad and M. Rangamma [424] have obtained certain results on the form of solutions of Lehmer’s problem (165). Let s be the number of primes of the form p ≡ 2 (mod 3) in the prime factorization of n.If3 n, then n must have the form 214 · 3 · m + 1or214 · 3 · m + 65537, according as s is even, or odd. (192) P. Hagis, Jr. [173] further reﬁned Kishore’s result (174) that for any k > 2, ω(n) ≥ 1991 and n > 108171.(193) Estimates (182) are further improved to 3|n ⇒ ω(n) ≥ 298848, and n > 101937042 (194) In an unpublished manuscript, M. Deaconescu and J. Sandor´ [99] have shown that if a composite n has an odd number of prime factors p ≡ 3 (mod 4), then n cannot be a solution of (165). Recently, A. Grytczuk and M. Wojtowicz´ [163] have shown that if n is a compos- k/4 ite solution to (168) with least prime divisor p1 = 3, then ω(n) ≥ 3049 − 1509. If k/4 p1 > 3, then ω(n) ≥ 143 − 1. C. A. Nicol [318] proved that n ≥ 2isprime iff ϕ(n)|(n − 1) and (n + 1)|σ(n). We now give estimates for the solutions, or the number of solutions of the considered problems. Let n ∈ S be a solution of Lehmer’s problem (165). Then

~~r n < r 2 ,(195) where r = ω(n) (see C. Pomerance [334]). Subbarao and Siva Rama Prasad [445] have reﬁned this (on view of (180)) to: If n ∈ S∗ is a solution of problem (178), then~~

~~r n <(r − 1)2 −1 (196) Let N(x) = card{n ≤ x : n ∈ S}.Bysharpening a result of P. Erdos¨ [109], in 1977 Pomerance [334] derived: N(x) = O(x1/2(log x)3/4(log log x)−1/2). (197)~~

~~214 THE MANY FACETS OF EULER’S TOTIENT~~

Shan Zun [410] improved the exponent 3/4 to 1/2. (198) Let P denote the set of all odd primes. Grytczuk and Wojtowicz´ [163] have proved that for all m ≥ 2 there exists an inﬁnite subset P(m) of P such that: i) for every distinct primes p1,...,pm ∈ P(m) the square-free number n = p1 ...pm is not a solution of (168); ii) P(m) is maximal with respect to inclusion. For the corresponding set N ∗(x) = card{n ≤ x : n ∈ S∗} one has

N ∗(x) = O(x1/2(log x)2(log log x)−2); (199) see [445]. Based on a remark of A. Schinzel (see [171], p. 92) that for n = 2, p or 2p (where p is an odd prime) one has (ϕ(n) + 1)|n, and if the converse is true, G. L. Cohen and S. L. Segal [88] have studied the relation

~~(ϕ(n) + 1)|n.(200)~~

They proved the following theorem: If n satisﬁes (200), then one of the following assertions is true: a) n = 2, p,or2p (where p is an odd prime); b) n = mt, where m = 3, 4, or 6; (m, t) = 1 and t − 1 = 2ϕ(t); c) n = mt, where (m, t) = 1, ϕ(m) = j ≥ 4 and t − 1 = jϕ(t). Thus, in case b) one can write (according to (175)) ω(t) ≥ 14, while in case c) ω(t) ≥ 140. As a corollary, if Lehmer’s problem has no solutions, then n = 2, p, 2p are the only solutions of (200). If there are other solutions, too, then they have at least 15 distinct prime factors. (201) Let N1(x) = card{n ≤ x : n = 2, p, 2p and satisﬁes (200)}. Then, (see [88]),

~~1/2 3/4 −5/6 N1(x) = O(x (log x) (log log x) )(202)~~

~~Finally, we want to mention certain other analogs of Lehmer’s problem. R. L. Graham (see [171], p. 93) makes the following conjecture: For all k there are inﬁnitely many n such that~~

~~ϕ(n)|(n − k)(203)~~

~~This is true for k = 0, k = 2α (α ≥ 0) and k = 2α · 3β (α, β > 0). For ﬁxed k,C.Pomerance [335] has shown that the number of elements n ≤ x satisfying (203) is x O ; (204) log x~~

~~215 CHAPTER 3 in particular, for ﬁxed k, the numbers n have asymptotic density 0. He also showed that for each k, (203) has at least four solutions. V. Meally (see [171], p. 93) remarks that~~

~~ϕ(n)|(n + 1)(205) has at least two solutions. Are there inﬁnitely many? M. Deaconescu [95] proved the inequality~~

~~ϕ(n)(ϕ(n) − 1) ≥ (n − 1)S2(n), (206) where S2(n) is the case k = 2ofSchemmel’s totient ([391]) (see relations (26) of 3.1 and (131)), and with equality iff n = prime. Therefore~~

~~(n − 1)|ϕ(n)(ϕ(n) − 1) iff n = prime (207)~~

~~Deaconescu conjectures that for n ≥ 2~~

S2(n)|(ϕ(n) − 1) iff n = prime,(208) and expects that this to be as hard as Lehmer’s conjecture. For an extension of S2(n) with applications in group theory, see M. Deaconescu and H. K. Du [96]. Let n ≥ 3. Then (ϕ(n))2|(n2 + 1).(209) is impossible, since ϕ(n) is even, and −1isnot a quadratic residue mod4. On the other hand, if (ϕ(n))2|(n2 − 1), (210) one obtains a difﬁcult (but treatable) problem. By applying primitive divisors of Lucas sequences, related to a Pell equation, as well as a form of the Brun-Titchmarsh theorem (see [291], Chapter 8) due to H. L. Montgomery and R. C. Vaughan [294]. F. Luca and M. Krizekˇ [270] have recently proved that all solutions of (210) are n = 1, 2, 3. (211)

~~3.3 Equations involving Euler’s and related totients~~

~~1 Equations of type ϕ(x + k) = ϕ(x) Perhaps the oldest equation related to Euler’s totient is~~

~~ϕ(x) = ϕ(x + 1)(1)~~

~~216 THE MANY FACETS OF EULER’S TOTIENT~~

In 1917 R. Ratat (see [103] p. 140) noted that x = 1, 3, 15, 104 are solutions; a year later R. Goormaghtigh found the solutions x = 164, 194, 255, 495. All other solutions up to x ≤ 10000 are x = 584, 975, 2204, 2625, 2834, 3255, 3705, 5186, 5187 (see V. Klee [227] and L. Moser [302]). (2) In 1972 M. Lal and P. Gillar [236] used an IBM1620, Model1, to determine all solutions for x ≤ 105.In1978 M. Yorinaga [485] found 146 solutions for x ≤ 10928925, by using a computer HITAC20. R. Baillie [23] made extensive computations to obtain all solutions for x ≤ 108.Hefound 306 solutions of equation (1). (3) In 1999, S. W. Graham, J. J. Holt and C. Pomerance [156] by using C++ programming language to implement simple sieving and scanning procedures to compute the necessary ϕ-values and then look for solutions of (1), have obtained all solutions for x ≤ 1010. There are 1267 solutions. (4) It is known that ϕ(30n +1)>ϕ(30n) for all n ≤ 2·107.However, D. J. Newman [317] points out that this kind of numerical evidence may be extremely misleading. In fact, he proves that if a, b, c, d are nonnegative integers with a, c > 0 and ad − bc = 0, then there exists a positive integer n for which ϕ(a · n + b)<ϕ(c · n + d). The equation ϕ(x + 2) = ϕ(x), (5) has the solutions x = 4, 7, 8, 10, 26, 32, 70, 74 for x ≤ 100. There are 80 solutions satisfying x ≤ 10000. L. Moser [302] noted that if p and 2p −1 are both odd primes, then x = 2(2p − 1) is a solution. However, it is not known (and it is an extremely difficult problem) that there are infinitely many such primes. Lal and Gillard [236] obtained all solutions x ≤ 105,M.Yorinaga [485] those of x ≤ 4·106 (in total 7998), while for x ≤ 1010 there are in total a number of 7558421 solutions (see [156]). (6) The equation ϕ(x + 3) = ϕ(x)(7) has the curious property that has only two solutions for x ≤ 10000, namely x = 3, 5 (see A. Schinzel [394]). D. H. Lehmer extended the range to x ≤ 106 (see [171], p. 90). Finally, Graham, Holt and Pomerance considered x ≤ 1011, (8) without finding any new solutions. In 1956 W. Sierpinski [415] has shown that the equation ϕ(x + k) = ϕ(x)(9) has at least one solution for each value of k.In1958 A. Schinzel [394] showed that there are at least two solutions to (9) for all k ≤ 8 · 1047.(10) A year later, A. Schinzel and A. Wakulicz [396] have increased the bound to k ≤ 2 · 1058.(11)

~~217 CHAPTER 3~~

In fact, the following two propositions by Schinzel have been applied: Let k be odd. Then suppose that a sequence of primes 3 = p1 < p2 < ··· < pm satisﬁes the conditions i)(pi − 2)|p1 p2 ...pi−1 (2 ≤ i ≤ m), ∗ (12) ii)(pi − 1)| 2p1 p2 ...pi−1 (2 ≤ i ≤ m), where a|∗ b means that each prime factor of a also is a prime factor of b. Suppose k is not divisible by p1 p2 ...pm,andlet p j be the smallest prime in the product that does nor divide k. Then p k x = j (13) p j − k is a solution to (9). Let now k be even. Let q1, q2,...,qm be a sequence of primes such that a) 2q − 1isprime (1 ≤ i ≤ m), i (14) b) 2qi − 1 = q j (1 ≤ i, j ≤ m). Suppose k < q1q2 ...qm. Then there exists a prime q j in the sequence such that q j and 2q j − 1 both do not divide k, and

x = (2q j − 1)k (15) is a solution to equation (9). Using Mathematica, and the built-in function Prime Q, which test primality using a combination of the Miller-Rabin and Lucas tests, recently J. J. Holt [199] has raised the bound to k ≤ 5 · 1038926 is the result (10) (k = even). (16) There is a conjecture of L. E. Dickson [104] known as the ”prime k-tuple conjecture”. A special case of this conjecture is that there are infinitely many primes p with 2p − 1 prime. Therefore, this conjecture implies that for k even, there are infinitely many solutions to equation (9) (known as Schinzel’s conjecture). This remark (by Schinzel) appears also in M. Zhang [487]. There are also some solutions of particular type of the considered equation. D. Klarner [226] remarks that equation (1) has solutions x = 22n − 1 only for n = 0, 1, 2, 3, 4, 5. (17) A. Schinzel [394] shows that for all positive integers n, there exists a positive integer m such that the equation (9) for k = nm has at least two solutions. (18) Another method of finding solutions to equation (9) is due to Graham, Holt and Pomerance [156]: Suppose that k is even and that j and j + k have the same prime factors. Let g = ( j, j + k), and suppose that for a positive integer r, j j + k r + 1 and r + 1 (19) g g

218 THE MANY FACETS OF EULER’S TOTIENT are both primes that do not divide j. Then j + k x = j r + 1 (20) g is a solution of equation (9). It is known that, for a given k, there are only ﬁnitely many values of j such that j and j + k have the same prime factors (see e.g. R. Tijdeman [453], and the References therein). For example, for each even k ≤ 30 all possible values of j can be completely determined by elementary techniques (based on the study of certain exponential diophantine equations). Related to the asymptotic number of solutions of the considered equations, many conjectures have been stated. Based on the Hardy-Littlewood conjecture on the distribution of prime pairs (p, 2p − 1), p ≤ n (see [178]), M. Yorinaga conjectures that c n P(2, n) ∼ 2 · as n →∞,(21) 2 log2 n where P(k, n) denotes the number of solutions x ≤ n of the equation (9), and 1 c = 1 − = 0.6601618 ... is the twin-prime constant. His 2 ( − )2 p>2 prime p 1 conjecture is supported by numerical tables. Without any assumption, Graham, Holt and Pomerance [156] prove the following result. Let P1(k, n) be the number of solutions x ≤ n of equation (9), which are not in the form (20). In particular, when k is odd, P1(k, n) = P(k, n).Then for every k, 1/3 there is some n0(k) such that if n ≥ n0(k),thenP1(k, n)2 p 2 2 log at log bt Now, as a corollary of theorem (22) one obtains. ∗ g ∗ p − 1 ∗ Let k be even, and let c(k) = , where runs over all j( j + k) p − 2 ∗ j such that j and j + k have the same prime factors, runs over all primes p > 2 such that p| jk( j + k)/g3,andg = ( j, j + k). Then 0 < c(k)<∞;and if conjecture (23) is true, then as n →∞ n P(k, n) ∼ 2c2c(k) (24) log2 n

~~219 CHAPTER 3~~

~~Let now P0(k, n) be the number of solutions x ≤ n of equation (9) in the form (20). Then P(k, n) = P0(k, n) + P1(k, n), and corollary (24) asserts that if (23) is true, then~~

P(k, n) ∼ P0(k, n) as n →∞ (25) for each ﬁxed even number k. The following theorem is surprising, since it shows that P(k, n) ∼ P0(k, n) or P(k, n) ∼ P0(k, n) are not k≤n k≤n k≤n, keven k≤n true: Let ∗∗ br 2 c = (26) ( + )( + )2γ( )γ ( ) 1≤a

~~∗∗ where in one has (b, ar + 1) = 1 and ar + 1, br + 1are both primes. Then c < ∞,(27) and P0(k, n) ∼ cn as n →∞ (28) k≤n~~

~~A generalization of equation (9) to arithmetic progressions is~~

~~ϕ(x) = ϕ(x + k) = ϕ(x + 2k) = ···=ϕ(x + qk)(29)~~

For k = 1, q = 1, as we have seen from relation (2), x = 5186 is a solution. No similar progression with difference q = 1isknown. Note that a solution with q ≥ 3 immediately implies that ϕ(x) = ϕ(x + 3),soby(8), we must have x > 1011.(30) P. Erdos¨ [111] has conjectured that for k = 1 and arbitrary q,equation (29) is solvable. By numerical veriﬁcations many progressions of length 3 (i.e., q = 3) can be found. For example, there is exactly one progression of length 3, when k ∈{1, 2, 4, 5, 8, 11, 14, 23, 25, 26, 28, 29, 31, 37, 38, 41, 46, 47, 52, 53, 55, 56, 58, 59, 62, 67, 71, 73, 74, 76, 79, 80, 85, 86, 89, 92, 94, 97, 98} (31) There are exactly two progressions of length 3, when k ∈{16, 17, 22, 32, 34, 43, 44, 61, 82, 83, 88}.(32) There are exactly three progressions of length 3, when k ∈{64, 68}.(33) The ﬁrst progression of length 6 has k = 30; x = 583200 (see also the Sloane encyclopedia [429], sequence A050518). An extension of theorem (20) for the equation (29) can be stated as follows (see [156]):

~~220 THE MANY FACETS OF EULER’S TOTIENT~~

Suppose that j, j + k,..., j + qk all have the same prime factors. Deﬁne B = j( j + k)...(j + qk).Fori = 0,...,q deﬁne B b = , g = gcd(b , b ,...,b ), i j + ik 0 1 q (34) b B a = i = i g ( j + ik)g Suppose that for some positive integer r,

a0r + 1, a1r + 1,...,aqr + 1 (35) are all primes that do not divide j. Then Br x = j(a r + 1) = + j (36) 0 g is a solution to (29). As a corollary one obtains that, if one assumes the conjecture (a particular case of Dickson’s conjecture, or Schinzel’s Hypothesis H [395]) that for distinct positive integers a0, a1,...,aq , the integers (35) are all primes for inﬁnitely many r, then for any positive integer q there exists a positive integer k such that (29) has inﬁnitely many solutions. An arithmetic progression of length 10 with equal phi-values is the following (see [156]): ϕ(502781177052210 + 210i), 0 ≤ i ≤ 9 (37)

2 ϕ(x + k) = 2ϕ(x + k) = ϕ(x) + ϕ(k) and related equations A similar equation to the above considered ones is ϕ(x + k) = 2ϕ(x)(38) A. Makowski [279] has shown that for each positive integer k has at least a solution. Indeed, if (k, 6) = 1, let x = 2k;ifk is even, i.e. k = 2ab (a ≥ 1, b odd), put x = 2ab;ifk is odd and divisible by 3, i.e. k = 6s + 3, let x = 2s + 1. Therefore, (38) is solvable for any k.(39) J. Sandor´ [368], as well as L. Cseh and I. Merenyi´ [92] have studied the equation ϕ(x + k) = 3ϕ(x)(40)

~~The density of positive integers k for which (40) has at least a solution is at least 1/2; (41) in fact D. B. Tyler [461] stated that the density is at least~~

~~0.864 (42)~~

221 CHAPTER 3 butnodoubt, that the equation is solvable for all k,except k = 2. This seems to be still open. Particular solutions for 1 ≤ k ≤ 10 have been analyzed by Sandor´ [368] and recently by J. Sandor´ and L. Kovacs´ [381]. For k = 2 there are no solutions up to x ≤ 106, while for k = 1 there are 42, for k = 3there are 18, for k = 5 there are 83, for k = 7 there are 63 solutions up to x ≤ 106.Fork = 4the single solution is x = 3; for k = 6 the single solution is x = 3, for k = 8 there are two solutions: x = 5, 6; and for k = 10, x = 4, 9(where x ≤ 106). A. Makowski [280], [281] has studied the equation ϕ(x + k) = ϕ(x) + k (43) Since ϕ(x) ≡ 0(mod 2) for x ≥3, it easily follows that for k = odd, equation (43) is solvable only when k + 2isprime, in which case the only solution is x = 2. (44) For k = 2a or k = 3 · 2a (a ≥ 1) one may take x = 3 · 2a and x = 7 · 3a,soin these cases the equation is solvable. (45) More generally (see [281]) if p and q = pa − p + 1 are prime numbers, for k = (p − 1)pa one obtains the solution x = qps.(46) Therefore, the number of solutions of this form depends essentially on the number of known Mersenne primes (see Chapter 1 of this book). Hypothesis H of Schinzel ([395]) states the following: Suppose there are given s irreducible polynomials f1(x), f2(x), ..., fs(x) with integer coefficients, positive for large values of x and such that there is no integer > 1 which is a divisor of f1(x) f2(x)... fs(x) for every x. Then there exist infinitely many x such that all of f1(x), f2(x), ..., fs(x) are prime numbers. (47) By assuming (47), Makoswski shows that for all even k, equation (43) has infinitely many solutions. (48) For k = 2 the equation (43) is particularly interesting, since all x with (x, x + 2) twin primes, is a solution. But there are also composite solutions, e.g. x = 12, 14, 20, 44. L. Moser [302] has shown that there are no composite odd solutions x < 10000. (49) Perhaps there are no other odd solutions x excepting when (x, x + 2) is a twin- pair. It is easy to see that if p > 2 and 2p + 1 are both primes, then x = 4p is a solution of the equation. Another open problem related to this question is due to A. Makowski ([414], p. 232): Has the system of equations ϕ(x + 2) = ϕ(x) + 2,σ(x + 2) = σ(x) + 2 (50) at least a composite solution? Another equation introduced by A. Makowski [282] is ϕ(x + k) = ϕ(x) + ϕ(k)(51)

~~222 THE MANY FACETS OF EULER’S TOTIENT~~

He showed that at least one solution exists if k is even, or 3 k,ork = a i 0 a1 ... as = 2 + > ( = , ) = mF0 F1 Fs , where Fi 2 1isaFermat number, ai 1 i 0 s , Fs+1 prime, and (m, 2F0 F1 ...Fs Fs+1) = 1. (52) Aparticularly interesting case is k = 3, when it is at least not known whether there is a solution (see [282]). (53) In [171] (p. 91) one can read that J. Browkin showed that for k = 3 there is no solution with x < 37182142. Recently J. Sandor´ and L. Kovacs´ [381] have veriﬁed this for x ≤ 3 · 108. P. Jones [214] has considered the case k = 3, too. She showed that any solution, if exists, must be of the form x = 2pα or x = 2pα − 3, where p > 3isaprime. (54) Further, if x is a solution, then (i) x or x + 3 has at least 33 distinct prime factors, or (ii) x = 2pα (α odd), p ≡ 2 (mod 3), x > 1011 and x + 3 has at least 9 distinct prime factors. (55) For the general equation (51), Jones has proved that for k = 3m odd it has a solution, in the following cases: a) pα k, pβ = q − 2,α>βand q k; b) p k, p = 3q − 4, and q k; (56) c) p k, p = 9q − 16, and q k; d) p k, p = 3αq − 2ar, 3α − 1 = 2a−1(r + 1), q k, and r k. Finally, if 2m + 1 = 3αn, where (3, n) = 1, α ≥ 0; and if we suppose there exists a positive integer j such that j − ϕ(j) = n and 3α j − 2m+1 = p, then if k = 3pv, with (3v,2pj) = 1, the equation (51) has at least a solution. (57) A more general context of the above equation is the theory of so called ϕ-partitions introduced in 1991 by P. Jones [215]. A partition n = a1 + a2 + ···+ai of n is said to be a ϕ-partition if

~~ϕ(n) = ϕ(a1) + ϕ(a2) + ···+ϕ(ai )(58)~~

In the above paper there are studied characterizations of positive integers which have at least one ϕ-partition and those which have exactly one ϕ-partition, the number of ϕ-partitions of prime numbers, as well as techniques for constructing ϕ-partitions and reduced ϕ-partitions for various types of integers. A ϕ-partition is said to be reduced if each summand has exactly one ϕ-partition. In 1996 C. Powell [340] proved that Jones’ recursive algorithm gives a unique reduced ϕ-partition. Powell character- izes also the integers having exactly one reduced ϕ-partition. (59) A number x is called a Phibonacci number if satisﬁes the equation

~~ϕ(x) = ϕ(x − 1) + ϕ(x − 2), x ≥ 3 (60)~~

~~223 CHAPTER 3~~

A. Bager [22] raised the problem if there exists any composite Phibonacci number? The answer is yes, 1037 being the smallest one. P. J. Weinberger found 70 such numbers < 200, 000, 000. All known Phibonacci numbers are odd. An even Phi- bonacci number, if it exists, must exceed 101600. C. A. Nicol [320] proved that there exist inﬁnitely many numbers n such that ϕ(n) ≤ ϕ(n − k) for all 1 ≤ k ≤ n − 1, and inﬁnitely many m such that ϕ(m) ≥ ϕ(k) + ϕ(m − k) for 1 ≤ k ≤ m − 1. The equation of k variables

ϕ(x1) + ϕ(x2) + ···+ϕ(xk) = ϕ(x1x2 ...xk)(61) has been solved independently by J. Sandor´ [367] and F. Luca [261]. There are at most a ﬁnite number of solutions, which can be found step-by-step. For example, when 2 ≤ x1 ≤ x2 ≤ ···≤ xk, the inequality

~~ϕ(x1) + ···+ϕ(xk) ≤ ϕ(x1x2 ...xk)(62) holds except for k = x1 = 2 and x2 = odd; and becomes an equality only in the following three instances:~~

~~ϕ(2·2) = ϕ(2)+ϕ(2), ϕ(3·4) = ϕ(3)+ϕ(4), ϕ(2·2·3) = ϕ(2)+ϕ(2)+ϕ(3). (63)~~

Essentially, besides k = 1 and x1, and the above, all other solutions can be obtained by ”completing with 1”, i.e. whenever (x1,...,xl ) is not in the above class, since u = ϕ(n1 ...nl ) − (ϕ(n1) + ··· + ϕ(nl )) > 0, set k = l + u and xl+1 = ···= xl+u = 1. (64) The equation xϕ(x) = yϕ(y)(65) of two arguments has the only solutions x = y.For this well-known property see e.g. [323] (Problem 19 of section 2.4). Related to the function f (n) = nϕ(n)(n = 1, 2,...) in W. Janous [212] it is proved that if F(x) = card{n > 0: f (n) ≤ x2}, then F(x) 1 1 lim = 1 + √ − (66) x→∞ ( − ) x pprime p p 1 p

α β By the same method, K.-W. Lau [241] showed that if fa,b(n) = n (ϕ(n)) a+b (α, β ≥ 0, αβ = 0), then putting Fa,b = card{n > 0: fa,b(n) ≤ x }, one has F , (x) lim a b = {1 + (p − 1)−b/(a+b) · p−a/(a+b) − p−1} (67) x→∞ x p

~~224 THE MANY FACETS OF EULER’S TOTIENT~~

~~P. Erdos¨ and C. W. Anderson [123] propose that for a given positive integer a there are only a ﬁnite number of integers b, with (b, a) = 1, such that the equation~~

~~bϕ(x) = ax (68) x has solutions. R. B. Eggleton [106] states that ϕ(x) = for inﬁnitely many x,but 3 x ϕ(x) = for all x. 4~~

~~3 Equation ϕ(x) = k, Carmichael’s conjecture Another old equation with a considerable interest is~~

~~ϕ(x) = k (69)~~

In 1900 A. Pichler (see [103], pp. 134-135) noted that for k > 1 there is no solution, while for k = 2n has the solutions x = 2abc ... (a = 0, 1,...,n + 1) if v b = 22u + 1, c = 22 + 1 are distinct Fermat primes, and 2u + 2v + ··· = n or n − a + 1, according as a = 0ora > 0. For k = 2 · qn (q > 3, prime) the equation has no solutions if p = 2qn + 1isnot prime; if p is prime, then has two solutions: x = p, 2p.Forq = 3, p = prime, it has the additional solutions x = 3n+1,2· 3n+1. n n−s For k = 2q the equation has no solutions if no one of the numbers ps = 2 q + 1 (s = 0, 1,...,n − 1) is prime and q is not a prime of the form 2h + 1, h = 2l ≤ n; but if q is such a prime, or if at least one ps is prime, then the equation has solutions 2 n−h s of forms bq , with ϕ(b) = 2 ; resp. aps, with ϕ(a) = 2 . For k = 2qr (q, r primes), the equation has no solutions if p = 2qr + 1 = prime and r = 2q + 1. If p is a prime, but r = 2q + 1, the two solutions are x = p, 2p. If p = prime, but r = 2q + 1, the two solutions are x = r 2, 2r 2.Ifp = prime and r = 2q + 1, all four solutions occur. In 1907 R. D. Carmichael proved that (see [103], p. 137) for k = 2m, with m > 1 odd, x must have the forms x = pα or x = 2pα, where p is a prime of the form p ≡ 3 (mod 4). In 1908 he gave a method of solving equations like (69), based on the testing of the equation for each factor x of a deﬁnite function of k.Hesolved also the equation ϕ(pα) = ϕ(qβ ) ([68]). In the same year, A. Ranum (see [103], p. 137) discovered a method for ﬁnding solutions of (69) when there are known all solutions of the equation ϕ(x) = m with m < k. This can be summarized as follows. When x is a prime power solution, i.e. x = pn, one can have solutions only if k = pn−1(p −1).Letnow x = ab, (a, b) = 1, 1 < a < b.Then ϕ(a)ϕ(b) = k. Hence, either ϕ(a) and ϕ(b) are both even or ϕ(a) = 1,

225 CHAPTER 3 i.e. a = 2. In the latter case, b is odd, yielding two solutions ϕ(b) = ϕ(2b) = k.Inthe former case ϕ(b) and ϕ(a) are both even, so k is a product of two even integers. This gives the following procedure for obtaining new solutions. Write k = (2u)(2v) and solve the equations ϕ(a) = 2u, ϕ(b) = 2v.Weobtain new solutions by multiplying together relatively prime solutions for each of the two equations. For a discussion, with numerical examples, see N. S. Mendelsohn [288]. In 1950 H. Gupta [170] showed that for k = m! (m > 1) the equation is solvable. Namely, x = (m!)2/ϕ(m!) is an integer, and a solution to (69). See also P. Erdos¨ [112]. Similarly, for k = m! · 2m, the equation has the solution x = 4m(m!)2/ϕ(m! · 2m), proved by C. S. Venkataraman [468]. In 1977 A. Aigner [2] showed that if a > 1isasquarefree integer, then there exist infinitely many solutions x,whenk is of the form k = ab2 (where b ≥ 1), and also infinitely many solutions x = y2 (y = 1, 2,...), when k = ac2 (c ≥ 1). There are integers k such that (69) has exactly 2or3,etc. solutions. Let Vϕ be the valence function of ϕ, i.e. let Vϕ(k) denote the number of solutions x of equation (69). For example, Vϕ(1) = 2, Vϕ(2) = 3, Vϕ(4) = 4, Vϕ(10) = 2, Vϕ(14) = 0, Vϕ(20) = 5, Vϕ(22) = 2etc. H.-J. Kanold and W. Sierpinski (see P. Erdos¨ [113]) have proved that Vϕ(n) = 2 for infinitely many n, such are the integers n = 2 · 36k+1 (k = 1, 2,...). Similarly, Vϕ(n) = 3 for infinitely many n, due to A. Schinzel [392] (we may take n = 12 · 712k+1, k = 0, 1, 2,...). S. S. Pillai [331] was the first to prove that

lim sup Vϕ(n) =+∞ n→∞ and that for almost all n, Vϕ(n) = 0. This result was given by A. Schinzel [393] (see also [414], p. 233). P. Erdos¨ [113] proved that if Vϕ(n) = a, then there exist inﬁnitely many such integers n. In 1935 P. Erdos¨ [114] showed that there exists δ>0 such that

c Vϕ(m)>m for inﬁnitely many m, and in 1979 K. R.√ Wooldridge [484] proved that the above statement holds in fact for every δ<3 − 2 2 (by using estimates from Selberg’s upper bound sieve).

~~226 THE MANY FACETS OF EULER’S TOTIENT~~

C. Pomerance [336], by using certain improvements on average in the Brun- Titchmarsh theorem due to Hooley, together with Bombieri’s sieve, obtained the re- finement δ<0.55655 This implies 5/9 Vϕ(m)>m for infinitely many m. A. Balog in 1984 obtained δ<0.65 (see [349], p. 321). This has been further refined by J. B. Fridlander [144]. As a consequence of the above results one has 5/9 Vϕ(n) = c0x + (x ) n≤x where c0 = ζ(2)ζ(3)/ζ(6). P. T. Bateman [30] has shown 1/2 Vϕ(n) = c0x + O(x exp{−c1(log x log log x) }), n≤x √ where c1 < 1/ 2isarbitrary. C. Pomerance in [336] proves

Vϕ(n) ≤ n exp(−(1 + o(1)) log n log log log n/ log log n) and conjectures, that this is best possible. This if true, would imply log log log x Vϕ(n) = c0x + x exp −(1 + ε) log x n≤x log log x for every ε>0. H. Heilbronn and H. Davenport (see [113]) proved 1 2 (Vϕ(n)) →∞as x →∞ x n≤x 2 1+c and conjectured that (Vϕ(n)) x for some c > 0. n≤x P. Erdos¨ [113] conjectures that (92) is true for every c < 1. The value c = 1/9 appears in [336]. ∗ If Vϕ (k) denotes the number of squarefree solutions to (69), then

~~∗ (1+o(1)) log n log log log n/ log log n Vϕ (n) ≤ n/e ,(70) see [336].~~

~~227 CHAPTER 3~~

~~# Let Vϕ (k) be the number of 1 ≤ m ≤ k such that Vϕ(m)>0. P. Erdos¨ (see [349], p. 321) showed in 1935 and in 1945, respectively that~~

# n Vϕ (n)< ,(71) (log n)1−ε for all n > n(ε) (ε>0 arbitrary, but ﬁxed) ( )k # n log log n Vϕ (n)> (72) log n for every k > 0 and sufﬁciently large n. These results were further improved by P. Erdos¨ and R. R. Hall [126]. H. Maier and C. Pomerance [277] deduced

# n 2 Vϕ (n) = exp{C + o(1)(log log n) } (73) log n where C = 0.81781465. K. Ford [138] further improved this to # n 2 Vϕ (n) = exp C(log n − log n) + D log n− log n 3 4 3 1 − D + − 2C log n + O(1) (73) 2 4 = . ... where logk n denotes the kth iterate of the logarithm of n,andD 2 17696874 K. B. Stolarsky and S. Greenbaum [441] denoted by L(k) the l.c.m. of the solutions of equations (69). If G(k) = gcd of all numbers ak = 1(1≤ a ≤ L(k), G(k) (a, L(k)) = 1), then they proved that the ratio can be arbitrarily large. L(k) In 1907 R. D. Carmichael (see [103], p. 137) proposed as an exercise to prove that for every n ≥ 1itispossible to ﬁnd an m such that

~~ϕ(n) = ϕ(m), i.e. Vϕ(n) = 1forall n.(74)~~

Forafew years it was thought that Carmichael had proved this. (74) is the famous Carmichael conjecture. 37 Already in 1922 Carmichael [67] showed that if Vϕ(n) = 1, then n > 10 .(75) In 1947 V. L. Klee [228] showed n > 10400, while P. Masai and A. Valette [285] in 1982 raised the bound to n > 1010000.(76) In 1994, still based essentially on Klee’s method, A. Schlaﬂy and S. Wagon [397] improved this to n > 10107 .(77) K. Ford [138], [139] showed that if there is a counterexample to Carmichael’s conjecture, then a positive proportion of totients are counterexamples. He improved (77) to 101010 .(78)

~~228 THE MANY FACETS OF EULER’S TOTIENT~~

Ford ([138], [139], [140]) has proved also a famous conjecture of Sierpinski, namely that every integer n ≥ 2isavalue of the function Vϕ (i.e. all integers ≥ 2 occur as multiplicities). (79) The corresponding conjecture (due also to Sierpinski) for the function σ (i.e. that for every n ≥ 0 there is a number m for which σ(x) = m has exactly n solutions) has been settled by K. Ford and S. Konyagin [142]. There are examples of even numbers n such that there is no odd number m such that ϕ(m) = ϕ(n).L.Foster (see [171], p. 94) has given n = 29 · 2572 as the least such number. (80) Some conditions for the validity of Carmichael’s conjecture are given in W. A. Ramadan-Jradi [343]. M. V. Subbarao and L.-W. Yip [447] have considered the equation

Sm(x) = k (81) ( ) ( ) where Sm n denotes the Schemmel totient function. Let VSm k denote the number of solutions of equation (81). They proved the following result: Let m ≥ 3beofthe form pα − 2, where p is an odd prime, and α ≥ 1. The H-Hypothesis by Schinzel implies that for any given integer n > 1, there exist inﬁnitely many integers k such ( ) = that VSm k n.(82) They conjecture that this is true for all m ≥ 1ﬁxed. (83) Let ϕ∗ be the unitary analogue of the totient function. M. Ismail and M. V. Sub- barao [206] consider the equation

~~ϕ∗(x) = k (84)~~

Since for k odd, (84) has a solution iff k = 2m − 1, the unitary analogue of the Carmichael’s conjecture is clearly false. However, for k = even, there are some evidences that the conjecture could be true (see [206]). As we have seen, equation (69) is not solvable for odd k > 1. But this may be true for even values of k, too. In 1956 A. Schinzel [392] showed that for every a ≥ 1 and k = 2 · 7a, (69) is not solvable. (85) More generally, for every positive integer s there exists a positive integer k divisible by s such that the equation ϕ(x) = k has no solutions. (86) D. Lind [258] proved that if m is a prime such that 2m + 1iscomposite, then ϕ(x) = 2m has no solutions. (87) Since, by Dirichlet’s theorem on arithmetic progressions, there are inﬁnitely many primes of the form 3n + 1, and 2(3n + 1) + 1 = 3(2n + 1) is composite, there are an inﬁnite number of primes satisfying theorem (87). In 1963 J. L. Selfridge [406], and also P. T. Bateman [31] gave the solution of a problem proposed by O. Ore:

~~229 CHAPTER 3~~

a For every a ≥ 1 there exists an odd integer ka such that for k = ka · 2 , equation (69) is not solvable. (88) a (i.e. ka · 2 is not a value of Euler’s function, or it is a ”nontotient”). In 1976 N. S. Mendelsohn [288] proved that there exist inﬁnitely many primes p such that for every a ≥ 1, p · 2a is a nontotient. (89) More general theorems have been obtained in 1989 by K. Spyropoulos [436]. Forexample, let p1,...,pm be distinct odd primes, and let a(1), ...,a(m) be arbitrary positive integers. Suppose that pi ≡ 1 (mod L), i = 1, 2,...,m, where = { + , + 2,..., + n} n a(1) ... a(m) L lcm 1 2 1 2 1 2 .Then 2 p1 pm is a nontotient. (90) MZhang [488] proves that a nontotient can have an arbitrary divisor, and gives two classes of odd numbers such that for the odd number k of the ﬁrst class 2αk is a nontotient for a given positive integer α, while for the odd number k of the second class, 2αk is a nontotient for arbitrary positive integer k.(91) ( ) (k) Am n Finally, if Am(n) = card{x ≤ n : ϕ (x) = m for some k}, then lim is n→∞ n equal to 1 if m = 2s, and to 0 otherwise. See E. Jacobson [209].

4 Equations involving ϕ and other arithmetic functions There are many equations on Euler’s or related totient, connected to other arithmetic functions. In 1894 A. P. Minin (see [103], p. 313) proved that all solutions of

~~ϕ(x) = d(x)(92) are x = 1, 3, 8, 10, 18, 24 and 30. In fact~~

ϕ(n) ≥ d(n) for n odd, with equality only for n = 1, 3; (93) and ϕ(n)>d(n) for all n > 30,(94) se e.g. [363], pp. 110-111. By answering a problem by Jankowska, P. Erdos¨ [115] proved that the system of equations of two arguments

~~ϕ(x) = ϕ(y), d(x) = d(y), σ (x) = σ(y)(95) has inﬁnitely many solutions. More generally, for every k there are k (squarefree) integers x1,...,xk satisfying~~

~~ϕ(xi ) = ϕ(x j ), d(xi ) = d(x j ), σ (xi ) = σ(x j ), 1 ≤ i < j < k (96)~~

~~230 THE MANY FACETS OF EULER’S TOTIENT~~

~~Further, he proves (see [116]) that for every k there are squarefree integers x1, x2,...,xk satisfying~~

~~(xi , x j ) = 1,ϕ(xi ) = ϕ(x j ), d(xi ) = d(x j ), 1 ≤ i < j ≤ k (97)~~

The same result holds, when we replace ϕ(n) by σ(n). For further properties of related nature, see P. Erdos,¨ K.Gyory¨ and Z. Papp [125]. The equation ϕ(x)d(x) = ϕ(x) + x − 1 (98) has the general solutions x = 1, or x = prime; and in fact

~~ϕ(n)d(n) ≥ ϕ(n) + n − 1forall n ≥ 1 (99)~~

(see [363], p. 112), improving ϕ(n)d(n) ≥ n, due to R. Sivaramakrishnan. All solutions of ϕ(x)(ω(x) + 1) = x (100) are given by x ∈{1}∪{2a · 3b : a > 0, b ≥ 0} (see [363], p. 112), and in fact

~~ϕ(n)(ω(x) + 1) ≥ n for all n ≥ 1 (101)~~

~~The equation σ(x) = ϕ(x) + d(x)(x − ϕ(x)) (102) has the solutions x = 1, or x = prime. In fact,~~

σ(n) ≤ ϕ(n) + d(n)(n − ϕ(n)) for all n ≥ 1,(103) improving nd(n) ≥ ϕ(n) + σ(n), due to C. A. Nicol (see [363], p. 114). Let ψ be the Dedekind arithmetical function, given by ψ(n) = n (1 + 1/p). p|n, pprime The equation ψ(x) = ϕ(x) + 2ω(x) (104) has x = 1orx = prime, as general solutions. In fact, ψ(n) ≥ ϕ(n)+2ω(n), improving (by ψ(n) ≥ σ(n))relation σ(n) ≥ ϕ(n)+2ω(n), due to C. A. Nicol (see [363], p. 196). All solutions of ψ(x) = 3ω(x)ϕ(x)(105) are x = 1and 2. Indeed, for n even one has ψ(n) ≤ 3ω(n)ϕ(n) with equality only for n = 1orn = prime, while for n odd one can write ψ(n) ≤ 2ω(n)ϕ(n) (see [363], p. 195) so (105) is impossible for x odd, x = 1.

~~231 CHAPTER 3~~

~~All solutions of σ(x) + ϕ(x) = x · 2ω(x) (106) are x = 1orx = prime, see C. A. Nicol [319]. Let k > 1beaﬁxed integer. The equation~~

σ(x) + ϕ(x) = kx (107) has been studied by C. A. Nicol, too. He proved that all solutions must satisfy log(k − 1) ω(x)> (108) log 2 and conjectured that for k > 2, all solutions are even. If k is odd, then x is even, or the square of an odd composite number. For k > 2, x cannot be squarefree. (109) For k = 3, a particular solution of equation (107) is given by x = 2α · 3 · q,ifone assumes that q = 7 · 2α−2 − 1isaprime (α > 2).(110) In 1971 S. A. Sergusov [408] proved that √ ϕ(x) = x + 1 − 2 x + 1 (111) iff x = p · q, where p < q are twin primes (i.e. p, q are primes with q = p + 2). A similar result is true for √ σ(x) = x + 1 + 2 x + 1 (112) W. G. Leavitt and A. A. Mullin [242] proved that the solutions of

(x − 1)2 − σ(x)ϕ(x) = 4 (113) are the products of two twin primes. In fact, if x = pq, with p − q = m (p, q primes), then x is a solution of the more general equation (x − 1)2 − σ(x)ϕ(x) = m2 (114) For particular m,however there are also solutions of other type. For example, when m = pk − 1, with p and 2p − 1 being primes, x = pk(2p − 1) is a solution of (114). The equation doesn’t have solutions of type x = pqr for r ≥ 2, and for any m (p < q primes). (115) The complete study of equation (114) remains however, open. Another interesting equation is ([371])

~~σ(x) = ϕ(x)d(x)(116)~~

~~In 1988 J. Sandor´ [369] (see p. 11) proved that~~

~~σ(n) ≤ ϕ(n)d(n) for all n odd (117)~~

232 THE MANY FACETS OF EULER’S TOTIENT with equality only for n = 1, 3(see also [363], p. 211, and [370], p. 65). Therefore the odd solutions are x = 1, 3. Equation (116) is included also in D. Wells [480], p. 127, where it is mentioned that the least three solutions are x = 1, 3, 14. The only even solutions of type x = 14k, where k is odd and 7 k are x = 14 and 42 (see [371]). (118) Sandor´ proves also that there are no even solutions x with 3 x and ω(x) ≥ 3; and also with 6|x and ω(x) ≥ 4. (119) All even solutions up to x ≤ 105 are x = 14 and 42. A further reﬁnement of (117) can be found in [372]. Let ϕ∗, d∗,σ∗ be the unitary analogues of the functions ϕ,d,σ. Note that, d(n) ≥ d∗(n) and σ(n) ≥ σ ∗(n),but ϕ(n) ≤ ϕ∗(n). Since σ(n) σ ∗(n) ≤ , with equality only for squarefree n,(120) d(n) d∗(n) and σ ∗(n) ≤ d∗(n)ϕ(n) for all n ≥ 3, odd,(121) with equality only for n = 3, we get σ(n) σ ∗(n) ≤ ≤ ϕ(n) for all n ≥ 3, odd (122) d(n) d∗(n) with equality only for n = 1and 3. Relation (120) is proved also in [373]. Results of type d∗(n) · n ≤ ϕ∗(n)(d∗(n))2 ≤ n2 (123) as well as additive variants, are included in [385]. The inequality √ σ(n) ( n − 1)2 √ ≥ + n,(124) d(n) d(n) with equality only if n is 1, p,orp2, where p is a prime, is due to E. S. Langford [239]. r r = ai > ( ) = If n pi is the prime factorization of n 1, let B n ai pi (see [291], i=1 i=1 pp. 143-147), B(1) = 0. Then the only solutions of the equations

~~ϕ(x)σ (x) + 1 = xB(x)(125) and ϕ(x) + σ(n) = x + B(x)(126) are x = p (prime), see K. T. Atanassov [17].~~

~~233 CHAPTER 3~~

~~Let pn be the nth prime. L. Moser [303] proposed the equation~~

~~1 + ϕ(1) + ϕ(2) + ···+ϕ(x) = px (127) and showed that x = 1, 2, 3, 4, 5, 6 are the only solutions. In fact~~

~~1 + ϕ(1) + ϕ(2) + ···+ϕ(n)>pn for n > 100 (128)~~

~~Afamous, unsolved equation is (see e.g. [171], p. 94)~~

~~ϕ(x) = σ(y)(129)~~

Since for p, q primes ϕ(p) = p − 1 and σ(q) = q + 1, clearly p − 1 = q + 1iff p p = q + 2, so any pair of twin primes is a solution. If Mp = 2 − 1isaMersenne p+1 p p+1 prime, then by ϕ(2 ) = 2 = σ(Mp), x = 2 , y = Mp is a solution pair of (129). On the other hand, it is not known if there exist inﬁnitely many twin primes, or inﬁnitely many Mersenne primes.

5 The composition of ϕ and other arithmetic functions In what follows we shall study equations, and related inequalities, for the composition of ϕ and other functions or sequences. In a paper by L. Alaoglu and P. Erdos¨ [3] from 1944 one can read that P. Poulet gave many solutions to the equation

~~ϕ(σ(x)) = x (130)~~

~~For x ≤ 2500 all solutions are x = 1, 2, 8, 12, 128, 240, 720. Two further solutions are x = 215 and 231. They note also that it can be shown that for every c > 0,~~

σ(ϕ(n)) ≥ cn (131) except for a set of density zero. Based on numerical investigations and particular cases by K. Kuhn, A. Makowski 1 and A. Schinzel [283] conjecture that (131) holds true for all n, with c = : 2 n σ(ϕ(n)) ≥ for all n ≥ 1 (132) 2 Kuhn showed that this is true for all n with ω(n) ≤ 6. Further, she found the solutions x = 22i +1 − 2 (0 ≤ i ≤ 5) of the equation x σ(ϕ(x)) = (133) 2

~~234 THE MANY FACETS OF EULER’S TOTIENT~~

After Makowski and Schinzel’s paper, inequality (132) was ﬁrst investigated in 1988 and 1989 by J. Sandor´ [374], [375], [376]. Similar problems, with stronger conjectures (σ replaced with ψ)were included also in [369] and [377]. In [376] the following lemma is proved: Let A denote the set of all numbers m with the property σ(ϕ(m)) ≥ m, m odd (134) Then for all even numbers n,having greatest odd divisor in A one has (132). Since for odd m one has ϕ(2m) = ϕ(m), this easily implies that (132) is true iff (134) is true. This has been rediscovered in 1997 by G. L. Cohen [85]. Let a ∧ b denote the property that there exists at least a prime divisor of a which doesn’t divide b. Let Js (s ≥ 1 integer) be the Jordan totient. The following is proved in [376]: Let S denote the set of all m > 1 such that

ks −(k−1)ω(m) σk(Js(m)) ≥ m · 2 (135) k (where k, s ≥ 1 are positive integers, and σk(a) = d ). d|a Let p denote a prime. Then, if m ∈ S and p|m, then mp ∈ S, too. If n ∈ S, p m, s and (p − 1) ∧ Js(m), then mp ∈ S, too. As a corollary one obtains a set B ⊂ S of odd numbers, for which inequality (135) is true. (Note that for k = s = 1, this inequality reduces to (134)). If n ≥ 2is an even number with greatest odd divisor m ∈ B, then nks σ (J (n)) ≥ (2s − 1)k · 2−(k−1)ω(n) (136) k s 2ks − k + 1 We conjecture that (135) holds true for all odd m ≥ 1, and (136) holds true for all even n ≥ 2. Inequality (136) reduces to (132) for k = s = 1. In Guy [171] one ﬁnds that Selfridge, Hoffman and Schroeppel found 24 solutions to equation (130). Cohen [85] notes that Terry Raines found 10 further solutions, and that he has found 8 more. There are all solutions up to 109.(137) In [429], sequence A001229 one ﬁnds that F. Helenius has found 365 solutions. Any solution of (130) gives a solution to

~~σ(ϕ(x)) = x (138)~~

Indeed, let x = σ(y), where y is a solution to (130). Then ϕ(x) = y,so σ(ϕ(x)) = σ(y) = x. Forfurther results on the Makowski-Schinzel conjecture we note that, by using Brun’s sieve, C. Pomerance [337] proved that σ(ϕ(n)) inf > 0 (139) n

~~235 CHAPTER 3~~

In 1992 M. Filaseta, S. W. Graham and C. Nicol [134] veriﬁed (132) for n = p1 p2 ...pk for k ≥ k0,where pi is the ith prime. U. Balakrishnan [24] proved the same for all squarefull integers n. More generally, if n is k-full (k ≥ 2), then σ(ϕ(n)) 1 ≥ (140) n ζ(k) Various classes of integers n for which (132) is true, are described also in [85]: 1◦ Any positive integer n of the form 2am, where i) the distinct prime factors of m are either Fermat primes or primes p ≡ 1 (mod 3), with at most eight of the latter; ii) m is a product of primes of the form 2br + 1 with b ≥ 1 and r prime (similar example is given in [375]); 2◦ Any positive integer n which is a product of primes less than 1780. (141) In [161] A. Grytczuk, F. Luca and M. Wojtowicz´ prove that the lower density of the set of integers satisfying the inequality is greater than 0.74. (142) In [162] they give various sufﬁcient conditions for the validity of inequality (132). > = a1 a2 ... ar ≤ Forexample, if the prime factorization of n 1isn p1 p2 pr , where 2 p1 ≤ ···≤ pr , and if p1 ≥ ω(n), then the inequality is true. (143) More generally, if r−1 1 ≤ − 1 + 2 ,() 1 r 144 i=1 pi 2 2pr then the inequality is true. Other results state that if n ≥ 2, and if 1 ≤ 1,() a 145 pa n p 2 then (132) is true. For n = m! (m ≥ 1),relation (132) is true, with equality only for m = 2or3. Recently, K. Ford [141] has shown that (131) is true with c = 1/39.4 for all n.Further results can be found in F. Luca and C. Pomerance [271]. Cohen [85] proves also that (134) holds true for m equal to a product of Fermat primes. For such a product, there is equality iff m = F0 F1 ...Fk for 0 ≤ k ≤ 4. (146) S. W. Golomb [150] remarks that if p > 3 and 2p − 1are primes, and m ∈ {2, 3, 8, 9, 15}, then x = (2q − 1)m is a solution to

~~ϕ(σ(x)) = ϕ(x)(147)~~

~~There are also other solutions, like x = 1, 3, 15, 45. Are there inﬁnitely many? He gives also the particular solutions x = 1, 87, 362, 1257, 1798, 5002 and 9374 to~~

~~236 THE MANY FACETS OF EULER’S TOTIENT the equation σ(ϕ(x)) = σ(x)(148) If p and (3p − 1)/2 are both primes, then x = 3p−1 are solutions to~~

~~σ(ϕ(x)) = ϕ(σ(x)) (149)~~

~~In fact, it can be proved that for all primes p ≥ 5,~~

~~σ(ϕ(p)) > p >ϕ(σ(p)) (150) and for all primes p,~~

~~ϕ(σ(2p−1)) > 2p−1 >σ(ϕ(2p−1)) (151)~~

~~For these, and related results, see H. Iwata [208]. Fortwo arithmetic functions f, g : N → N, put~~

~~ f,g(n) =|f (g(n)) − g( f (n))|, n = 1, 2,... (152)~~

~~J. Sandor´ [378] (see also [363], pp. 175-178) has proved that~~

~~lim sup σ,ϕ(n) =+∞ (153) n→∞ and conjectures that lim inf σ,ϕ(n) = 0 (154) n→∞ He proves also that~~

~~lim sup ,ϕ(n) =+∞, lim inf ,ϕ(n) = 0 (155) d →∞ d n→∞ n and remarks that x = 1 and x = 2p−1 (p ≥ 3prime) are solutions to~~

~~d(ϕ(x)) = ϕ(d(x)) (156) and asks for the general solutions. Let S(n) be the Smarandache function,deﬁned by S(n) = min{k ≥ 1: n|k!}. Without any assumptions, one has~~

~~lim sup ,ϕ(n) =+∞, lim inf ,ϕ(n) ≤ 1 (157) S →∞ S n→∞ n Now, x = p prime is a solution of the equation~~

~~S(ϕ(x)) = ϕ(S(x)) (158)~~

~~237 CHAPTER 3~~

What are the most general solutions of this equation? The equation σ(x) ϕ x = x (159) x (where [a] denotes the greatest positive integer ≤ a) has the single solution x = 1, since A. Oppenheim [326] showed that σ(n) ϕ n ≤ n for all n,(160) n with equality only for n = 1. A similar inequality is valid when σ is replaced with ψ (see [369], p. 7), so x = 1isagain the solution of the similar equation to (159), when σ is replaced with ψ. Also n ϕ n ≤ (ϕ(n))2,(161) d(n) with equality only for n = 1, see [379]. The equation ϕ(|xm − ym|) = 2n (162) where m, n are given positive integers and m ≥ 2 has been studied by F. Luca [262]. He first proves that it suffices to find those (x, y) for which x > y ≥ 1; (x, y) = 1, and m = 4. (163) All solutions satisfying (163) are given by

− − (x, y) = (22k 1 + 1, 22k 1 − 1), where k ≥ 1 and 22k + 1prime; (164) or (x, y) = (22k , 1) for k = 0, 1, 2, 3 The determination of all positive integers x, y, m, n of the equation ϕ(xm − ym) = xn + yn (165) is the main objective of another paper by F. Luca [263]. These solutions are given by (x, y, m, n) = (2k + 1, 2k − 1, 2, 1) k ≥ 1 integer (166) The equation ϕ(|xm + ym|) =|xn + yn| (167) where x, y are integers, and m, n positive integers, has been studied in [264]. It suf- ﬁces to consider x ≥ 0 and x ≥|y|.Asolution (x, y, m, n) of (167) is called trivial, if (k, 1 − k, 1, 1) for some positive integer k, or (x, y, m, n) = (168) (1, 0, m, n) for some positive integers m, n

~~238 THE MANY FACETS OF EULER’S TOTIENT~~

All nontrivial solutions (i.e. satisfying x ≥ 0, x ≥|y| and (168)) are given by   ( , , n + , n) n,  2 0 1 for some positive integer or ( , , + , ) , ( , , , ) = 2 2 n 1 n for some positive integer n or ( ) x y m n  ( , , + , ) , 169  3 3 n 1 n for some positive integer n or (3, 1, 2, 1)

~~It is not known (see [265]) if the equation~~

ϕ(5m − 1) = 5n − 1 (170) in positive integers m, n,has any solutions at all. The equation n ϕ = 2a (171) k n n has been considered in [266]. Since = ,itsufﬁces to suppose n ≥ 2k. k n − k Let A ={k ≥ 1: ϕ(k) is a power of 2} (172) Then all solutions of equation (171) for n ≥ 2k are the following: 1) k = 1 and n ∈ A;2)k = 2andn is either a Fermat prime, or n ∈{22, 2 · 3, 222 , 223 , 224 , 225 };3)k = 3 and n ∈{6, 10, 17, 18, 257, 65537}.(173) Remark. It is well known that n ∈ A iff n either is a power of 2, or n = α 2 p1 ...pt ,forsome α ≥ 0, t ≥ 1, where p1 < ··· < pt are Fermat primes (i.e. n = 1, 2orforn ≥ 3, the regular polygon with n sides can be constructed using only the ruler and compass), due to A. Cunningham from 1915 (see [103]. p. 140). Equations on the composition of Euler’s totient with Fibonacci or Lucas sequences appear in [267] and [268]. Forexample, in [267] it is proved that

ϕ(Fn) ≥ Fϕ(n) for all n ≥ 1,(174) with equality only for n = 1, 2, 3. A similar inequality is valid for d (with equality for n = 1, 2, 4), while for σ the sign of inequality is reversed (with equality for n = 1, 3). In [268] one can ﬁnd:

~~ϕ(Lm) = Ln (175) iff (m, n) = (0, 1), (1, 1), (2, 0), (3, 0) (where Ln+2 = Ln+1 + Ln, L0 = 2, L1 = 1 is the classical Lucas sequence);~~

~~ϕ(Fm) = Ln (176)~~

~~239 CHAPTER 3~~

iff (m, n) = (1, 1), (2, 1), (3, 1), (4, 0), (5, 3), (6, 3) (where F2 = Fn+1 + Fn, F0 = 0, F1 = 1isthe classical Fibonacci sequence). Let r and s be two non-zero integers with r 2 + 4s > 0. A binary recurrence sequence (un)n≥0 is a sequence such that u0 and u1 are integers and

~~un+2 = run+1 + sun, n ≥ 0~~

Let α, β denote the two roots of the equation x2 − rx − s = 0. It is well known n n that un = aα + bβ (n ≥ 0), where a, b are two constants. If (r, s) = 1, u0 = 0, u1 = 1, then (un) is called a Lucas sequence of the ﬁrst kind; while if (r, s) = 1, u0 = 2, u1 = r,then(un) is called a Lucas sequence of the second kind.In[269] the following is proved: Let (un) be a Lucas sequence of the ﬁrst kind such that its characteristic equation has real roots. Then ϕ(|un|) ≥|uϕ(n)| for all n ≥ 1 (177) with equality iff: i) n = 1; ii) n = 2 and |r|=1, 2; iii) n = 3, |r|=1, and s = 1. Abinary recurrent sequence is called nondegenerate if ab = 0. In [268] there are studied equations of type

~~ϕ(|aun|) =|bvn| (178) where (un), (vn) are certain binary recurrent sequences, and (un) is nondegenerate.~~

6Perfect totient numbers and related results As we have seen in 5. of 3.2, the iteration of Euler’s totient has some interesting congruence properties. In 1975 T. Venkataraman [469] deﬁned a perfect totient number n by

T (n) ≡ ϕ(n) + ϕ(2)(n) + ϕ(3)(n) + ···+ϕ(r)(n) = n (179) where r = r(n) is the smallest integer such that ϕ(r)(n) = 1. We note that the function r(n) was ﬁrst considered by S. S. Pillai, for its properties see e.g. [291], p. 35. Venkataraman proves that if n = 3 · (4 · 3m−1 + 1) with 4 · 3m−1 + 1 = prime, then n is a perfect totient number. (180) The study of PTN (for perfect totient numbers) however was initiated by L. P. Cacho [59], who in fact proved that 3p,for an odd prime p,isaPTN if and only if p = 4n + 1, where n is a PTN. Then Venkataraman’s result (180), follows as a corollary. A. L. Mohan and D. Suryanarayana [292] proved that 3p (p = odd prime) is not a PTN, if p ≡ 3 (mod 4).They found also sufﬁcient conditions on an odd prime p for 32 · p and 33 · p to be PTNs. For example, if b ≥ 0isaninteger, and

240 THE MANY FACETS OF EULER’S TOTIENT if q = 25 · 3b + 1, p = 2 · 32 · q + 1 are both prime, then 32 p is a PTN; and if q = 24 · 3b + 1 and p = 22 · q + 1are both prime, then 33 · p is PTN. (180’) In a recent paper, D. E. Iannucci, D. Moujie and G. L. Cohen [205] have obtained similar results. They gave also a table on all PTN less than 5 · 109, which are not powers of 3 (3k is PTN, as immediately follows). There are 30 such PTN. We quote the following results from [205]: If b ≥ 0 and q = 23 · 3b + 1 and p = 2q + 1are both prime, then 32 · p is PTN. If r = 22 · 3b + 1, q = 24 · r + 1, p = 22 · q + 1 are all prime, then 33 p is PTN. There are no PTNs of the form 3k p (k ≥ 4), where p = 2c · 3d · q + 1andq = 2a · 3b + 1 are primes with a, c ≥ 1 and b, d ≥ 0. (180”) Equation (179) has been considered also by D. L. Silverman [416]. P. Erdos¨ and M. V. Subbarao [130] note that T (n) = (1 + o(1))ϕ(n), for almost all n,(181) so that T (n) for inﬁnitely many n,(183) 2 and that T (2n) lim sup = 1 (184) n→∞ 2n Let F(x, c) = card{n ≤ x : T (n)>cn},where c > 0. Then for every 3 1 < c < we have for every t > 0 and ε>0, that 2 x x (log log x)t < F(x, 1 + c)< (185) log x (log x)1−ε for all x ≥ x0(c, t,ε). Further, we have x F(x, 1) = (c + o(1)) (186) log log log log x The details of proofs for (183)-(186) are not worked out in [130], but the authors state that they follow by the methods of this paper and that of [117]. 3 Erdos¨ and Subbarao conjecture that for 1 < c < c < , 1 2 2 F(x, 1 + c ) lim 1 =∞ (187) →∞ x F(x, 1 + c2)

241 CHAPTER 3 and that 3 x F x, = o (188) 2 log x Some other unanswered questions are: r(n) Does have a distribution function? (189) log n r(n) Does approach a limit for almost all n?(190) log n ∗ Let r ∗ = r ∗(n) be the smallest integer such that (ϕ∗)(r ) = 1, where ϕ∗ is the unitary analogue of Euler’s totient. It is not known if

~~r ∗(n) 0.(191)~~

~~M. Lal [235] computed for n ≤ 105 the maximum and minimum values of n for ∗ agivenr ∗ such that (ϕ∗)(r ) = 1. He obtained the bounds~~

~~log n < r ∗(n)<2.5 log n for n ≤ 105 (192)~~

~~3.4 The totatives (or totitives) of a number~~

1 Historical notes, congruences As we have mentioned in the introduction, in 1879, J. J. Sylvester called the positive integers r < n which are coprime with n the totatives (or ”totitives”, see e.g. [103], p. 124, or [194]). We note that theorem (126) of 3.2 gives all totatives of a number, which are prime numbers (except 1), a result due to S. Schatunowsky. In 1888 H. W. Lloyd Tanner (see [103], p. 131) studied the group G of the totatives of a number, ﬁnding all of its subgroups and the simple groups whose direct product is G. nϕ(n) It is easy to see that the sum of totatives of n is ,(1) 2 due to A. L. Crelle from 1845. Let t(n) denote the set of totatives of n, i.e. t(n) = {k :1≤ k < n,(k, n) = 1}. In 1850 A. Thacker introduced the function j φ j (n) = t , j ≥ 0,(2) t∈t(n) (i.e. the sum of jth powers of totatives of n), and noted that for j = 0itreduces to = a1 ... ar ( ) = Euler’s totient. Let n p1 pr be the prime factorization of n, and put s j m 1 j + 2 j + ···+m j . Thacker proved the following formula: φ ( ) = ( ) − j n + j j n − ... ( ) j n s j n p1 s j p1 p2 3 p1 p1 p2 p−1 p1,p2

~~242 THE MANY FACETS OF EULER’S TOTIENT~~

where the summation indices range over the combinations of p1, p2,... one, two,... at a time. Various authors, as J. Binet, J. Liuoville, E. Lucas, E. Cesaro,` L. Gegenbauer, etc. obtained other formulae, by using Bernoulli numbers or binomial coefﬁcients. W. Brennecke in 1852 proved that 1 γ(n) φ (n) = ϕ(n) n2 + (−1)ω(n) (4) 2 3 2 1 φ (n) = nϕ(n)[n2 + (−1)ω(n)γ(n)] (5) 3 4 where γ(n) = p1 p2 ...pr is the ”core” of n, while ω(n) denotes the number of distinct prime factors of n.For new proof, see [98]. J. Binet in 1851 proved that

~~φk(n) ≡ 0 (mod n) if k is odd; (6) and Mennesson showed in 1878 that 1 φ (n) ≡ ϕ(nk+1)(mod k), if k is odd (7) k 2 N. Nielsen proved in 1915 that~~

~~2 φ2k(n) ≡ 0 (mod n), φ2k+1(n) ≡ 0 (mod n )(8)~~

p − 3 where 1 ≤ k ≤ 1 , with p = p(n) = smallest prime divisor of n.J.Liouville 2 1 in 1857 proved the identity n k k k φk(d) = sk(n) = 1 + 2 + ···+n ,(9) d|n d which for k = 0 reduces to Gauss’ formula ϕ(d) = n. d|n In 1932 H. Davenport [94] rediscovered Thacker’s function and proved an asymptotic formula on it. For generalizations, in the regular convolutions setting, see L. Toth´ and P. Haukkanen [459]. In 1985 P. S. Bruckman [57] proposed for the Dirichlet series of φk(n) given by ∞ φ (n) f (s) = k k s the identity n=1 n

~~+ 1 k 1 k + 1 f (s) = + B + − ζ(s − j) k 1 ( + )ζ( − ) k 1 j k 1 s k j=1 j~~

~~243 CHAPTER 3~~

where Bm are Bernoulli numbers. For a generalization, see H. W. Gould and T. Shon- hiwa [153], where functions of type f (t) and f (t) are also 1≤t≤n,(t,a)=m 1≤t≤n,[t,a]=m studied. (Here (t, a) = gcd(t, a),[t, a] = lcm(t, a)). In 1889 C. Leudesdorf [253] introduced the function 1 ψ (n) = , j ≥ ( ) j j 0 10 t∈T (n) t which is another extension of Euler’s totient. He proved that for j = odd one has

1 2 1 ψ (n) = An − jnψ + (n)(11) j 2 2 j 1 for certain integer A.Asacorollary he deduced, that if n = psq, where q is not 2s divisible by the prime p > 3, then ψ j (n) ≡ 0 (mod p ).(12) 2 unless ( j, p) = 1 and (p −1)|( j +1).For example, ψ j (p) ≡ 0 (mod p ). Similarly, if p = 3, 2s ψ j (n) ≡ 0 (mod 3 ) if j is an odd multiple of 3; (13) and 2s−1 ψ j (n) ≡ 0 (mod 2 ) if p = 2, except when q = 1 (14)

For j = 1, by putting S = ψ1(n) one obtains (see also [179]) ≡ ( 2) , ; S 0 mod n if 2 n 3 n 1 S ≡ 0 (mod n2) if 2 n, 3|n; 3 1 (15) S ≡ 0 (mod n2) if 2|n, 3 n, n not a power of 2 2 1 S ≡ 0 (mod n2) if n = 2a 4

We note that the congruence S ≡ 0 (mod p2) (n = p = prime) was ﬁrst proved by J. Wolstenholme [483] (see also [367]). Extensions of Leudesdorf’s theorem were obtained e.g. by M. Rama Rao [345], I. Sh. Slavutskii [427]. See also [428]. For example, Slavutskii [427] proves the following:   n ( − pt−1)B ( n2) | j  1 t mod for 2  p|n ψ j (n) ≡ (15 )   t t−2 2  n (1 − p )Bt−1 (mod n ) otherwise 2 p|n

~~244 THE MANY FACETS OF EULER’S TOTIENT~~

l where t = (ϕ(p ) − 1)n (l ≥ 1 integer), and (n, 6) = 1. Here Bt denotes the t- x th Bernoulli number (the Bernoulli numbers (Bn) are generated by x/(e − 1) = ∞ n Bn x /n!, |x| < 2π, see Chapter V). n=0 By applying the von Staudt-Clausen theorem for Bernoulli numbers (see Chapter V), (15’) implies the following corollary: 1) If 2| j and (p, n) = 1for all prime numbers p such that (p − 1)| j, then

~~ψ j (n) ≡ 0 (mod n)~~

~~2) Let j be odd. a) If p − 1 does not divide j + 1for any prime p with p|n, then~~

~~2 ψ j (n) ≡ 0 (mod n )~~

~~b) If p| j for all primes p such that (p − 1)|( j + 1) and p|n, then~~

~~2 ψ j (n) ≡ 0 (mod n ).~~

~~Forapositive integer x,andt ∈ t(n), let fn(x) = (x − t)(16) t∈t(n)~~

~~By the classical Lagrange theorem one has~~

~~ϕ(n) fn(x) ≡ x − 1 (mod n), if n is prime (17)~~

~~However, for all n, the congruence (17) is not valid. In 1902 M. Bauer proved that if pa n (p prime), then~~

~~p−1 ϕ(n)/(p−1) a fn(x) ≡ (x − 1) (mod p )(18)~~

~~Particularly, p−1 pa−1 a f pa (x) ≡ (x − 1) (mod p )(19) Another result by M. Bauer states that for m > 2evenand2a n one has~~

~~2 ϕ(n)/2 a fn(x) ≡ (x − 1) (mod 2 )(20)~~

~~Particularly, 2 2a−2 a f2a (x) ≡ (x − 1) (mod 2 )(21) For proofs of Bauer’s theorems see Hardy-Wright [179].~~

~~245 CHAPTER 3~~

~~Recently A. Junod [217] has extended (18) as follows: If p is a prime and m, n ≥ 0are integers, and f (x) ∈ Zp[x]; ord p(m) ≥ 1, then n mn f (x − km) ≡ f (x) (mod Zp[x]) 0≤k~~

~~2 The distribution of totatives The study of distribution of totatives was initiated by D. H. Lehmer in 1955 [245]. Lehmer introduced the function ∗ φ(n; k, l) = 1, 0 ≤ l < k,(22) nl~~

~~2 Ak ={n : k |n or there exists a prime p|n with p ≡ 1 (mod k)}, ϕ(n) B ={n : φ(n; k, l) = for 0 ≤ l < k}, (23) k k Ck ={n : k|ϕ(n)}~~

~~It is clear that Ak ⊂ Ck, Bk ⊂ Ck, and Lehmer proved that~~

~~Ak ⊂ Bk ⊂ Ck (24)~~

~~It is not difﬁcult to prove that C p ⊂ A p for a prime p,so~~

~~A p = Bp = C p (25)~~

~~In 1957 P. J. McCarthy [70] proved that~~

Ak = Bk if k is not squarefree,(26) and asked if the result could be extended to all composite k. This was done by P. Erdos¨ [118] who proved that the set Bk \ Ak is inﬁnite for all composite k, B2p = C2p for an odd prime p, and Bk = Ck for k = p, 2p.(27)

~~246 THE MANY FACETS OF EULER’S TOTIENT~~

Some parts of the proof given by Erdos¨ have to be corrected; this has been done recently by R. R. Hall and P. Shiu [174]. Foraﬁxed modulus k, Hall and Shiu write x < y (mod k) to mean that the least non-negative residue congruent to x is less than that congruent to y. Then they prove the following theorem: Let a, b, c be positive integers which are distinct (mod k) and satisfying

~~(ab, k) = 1, c ≡ 0 (mod k), a + b ≡ c (mod k)(28)~~

Then there exists x such that ax < cx < bx (mod k). As a corollary, they obtain the proof of a conjecture by Erdos:¨ Let p, q be distinct odd primes such that pq ∈ Ak and pq ≡−1 (mod k). Then pq ∈ Bk.(29) The Lehmer, McCarthy, and Erdos¨ theorems, along with numerical examples, and particular cases, are presented also in the book by L. Holzer [200] (with other notations). Let t(n) ={t1 < t2 < ···< tϕ(n)}.In1940 Erdos¨ [119] conjectured that for some positive c one has 2 2 (ti+1 − ti ) < cn /ϕ(n)(30) where the sum is taken for 1 ≤ i ≤ ϕ(n) − 1. In 1962 C. Hooley [204] showed that ≤ α< for 1 2, α α−1 (ti+1 − ti ) n(n/ϕ(n)) (31) and 2 2 (ti+1 − ti ) n(log log n) (32)

M. Hausman and H. N. Shapiro [195] obtained the bound n2 log p c , ( ) ϕ( ) max 1 − 32 n p|n p 1 in the right side of (30). R. C. Vaughan [465] established the conjecture ”on the average”, and finally in 1986, H. L. Montgomery and R. C. Vaughan [295] completely proved (30). E. Jacobsthal [210] in 1960 defined a function g(n) as the least positive integer so that among g(n) consecutive positive integers there is at least a totient of n.For results on g(n),see [291], p. 34. Another function by Jacobsthal is J(n) = max{ti+1 − ti : i = 1,...,ϕ(n)−1}.There are not known nontrivial bounds on J(n). Erdos¨ asked if, for infinitely many x, there are two integers a and b, where a < b < x and (a, b) = 1, such that min{J(a), J(b)} > ln x.(33)

~~247 CHAPTER 3~~

~~3 Adding totatives~~

In 1976 B. R. Santos [388] considered the problem of finding the largest integer with the property that, when added to each of its totatives, yields a prime number. He proved that a largest such integer does exist, and conjectured that 12 was actually this largest integer. In 1978 M. Hausman and H. N. Shapiro [194] proved that the Santos conjecture is correct, and considered similar and more general questions. First they prove that only for n = 1, 2, 4, 6, 10, 12, 18 does there exist a value of k for which kn plus each totative of n is prime. For n = 1, 2, 4, 6, 10, 12, the smallest such k is k = 1, and for n = 18, the smallest such k is k = 892. (34) A similar question (raised in [388]) is of adding prime totatives to n either to yield only primes, or only composites. Hausman and Shapiro answer this questions in the following way: There exist infinitely many integers n to which each of its prime totatives can be added to yield a composite. (35) There is a largest integer n with the property that for some positive integer k each of the prime totatives of n added to kn always is a prime. (36) For adding composite totatives to n either to yield only primes or only composites, one has: There is a largest integer n with the property that for some positive integer k each of its composite totatives added to kn always is a prime. (37) There exists a largest integer n to which each of its composite totatives can be added to obtain a composite. However, in the general case, there exist infinitely many integers n for which there exists a positive integer k such that kn added to each of the composite totatives of n is composite. (38) The proofs of (34) and (35) are elementary, and based on a simple consequence of theorem (126) from 3.2 of Schatunowsky. Namely, let p∗(n) be the smallest prime totative of n. Then, √ p∗(n)< n for n > 30 (39)

~~Theorems (36) and (37) are based on more advanced results of Number theory, as the Siegel-Walﬁsz theorem; as well as an improvement of (39) for large values of n: p∗(n) 0, constant)(40)~~

~~In fact, the arithmetical function p∗ was introduced by B. R. Srinivasan [437], who proved that p∗(n) lim sup = 1,(41) n→∞ log n~~

248 THE MANY FACETS OF EULER’S TOTIENT and p∗(n) ∼ Ax,(42) n≤x ∞ p + − p A = p + r 1 r p r where 1 ... , and r is the th prime. r=1 p1 p2 pr In 1969 Y. Sankaran [387] improved (42) to p∗(n) = Ax + O(log x)(43) n≤x

4 Adding units (mod n) When n ≥ 2, for the set of totatives of n, t ∈ t(n), the classes t in the ring Z Z∗ n of residue classes modulo n,form a multiplicative group n,called also as the group of units of the ring Zn.If1≤ k ≤ n − 1, one can ask how many solutions ( , ) ∈ Z∗ × Z∗ t t n n there are, such that

~~ t + t = k (44)~~

Let s(k) be the number of solutions to (44). Then M. Deaconescu [95] proves that s(k) is given by ϕ(n) s(k) = ψ(d, n)(45) ϕ(n/d) where d = (k, n), and ψ(d, n) is the number of those automorphisms of the additive group of Zn,having exactly d ﬁxed points (i.e. with the property f (k) = k for such r (Z , +) = ai an automorphism f of n ). In [96] it is proved that if n pi is the prime i=1 r = bi ( ≤ ≤ ) factorization of n, and d pi is a divisor of n 0 bi ai , then i=1 − − − ψ( , ) = ai bi 1( − ) a j 1( − )() d n pi pi 1 p j p j 2 46 | / | / pi n d p j n d p |d i p j d

A corollary of (45) and (46) is the following proposition: 1) If n ≥ 2isodd, then every element of Zn is a sum of two units (i.e. (44) has always solutions). (47) 2) If n ≥ 2even, then k ∈ Zn is a sum of two units if and only if k is even.

~~249 CHAPTER 3~~

5 Distribution of inverses (mod n) For each t ∈ t(n) there exists a unique totative t, which is the multiplicative inverse of t (i.e. t · t ≡ 1 (mod n)). Then Z. Zheng in 1993 [493] proved that when n is odd, 1 √ card{t ∈ t(n) : t + t ≡ 1 (mod 2)}= ϕ(n) + O( n · d(n) log2 n)(48) 2 The proof makes use of estimates for character and Kloosterman-sums. When n is a prime power, or a product of two distinct primes, similar results have been deduced by W. Zhang [489]. In the same year, he settled also the general case (i.e. n = odd), too, see [490]. A generalization of (48) is given in [494]: m ϕ(n) √ card{t ∈ t(n) : t +t ≡ 1 (mod 2), t ≤ m}= · +O(d(n) n·log2 n)(49) 2 n where n is odd, and 1 ≤ m < n is a ﬁxed integer. For m = n − 1, one reobtains (48). Another generalization of (48) has been obtained by I. Z. Ruzsa and A. Schinzel [360]: Forevery choice of ε = 0, 1andd =−1, +1 and odd n we have t card t ∈ t(n) : t + t ≡ ε(mod 2), = δ = n ϕ( ) √ n ω(n) 2 = c ,δ + O(2 n · log n)(50) 4 n t where c ,δ = 1 + δ if n is a perfect square; and =1, otherwise. Here denotes a n n Legendre symbol, and ω(n) is the number of distinct prime factors of n. Let n > 2bearbitrary and 0 <δ≤ 1beﬁxed. Then √ card{t ∈ t(n) : |t − t| <δn}=δ(2 − δ)ϕ(n) + O(d2(n) n · log3 n)(51)

~~This is due to W. Zhang [491]. In [492] W. Zhang shows that for n > 2,~~

~~ϕ(n)n2k (t − t)2k = + O( kd2(n)n(4k+1)/2 2 n)() ( + )( + ) 4 log 52 t∈t(n) 2k 1 k 1 where k ≥ 1isaﬁxed positive integer.~~

~~250 THE MANY FACETS OF EULER’S TOTIENT~~

Let n > 2beanodd integer such that there exists a primitive root mod n. Let A denote the set of all primitive roots modulo n which are < n. Then t ∈ A ⇒ t ∈ t(n),and ∗ ϕ(ϕ(n))n2k (t − t)2k = + O( kd2(n)n2k+3/4 2 n)() ( + )( + ) 4 log 53 t∈A 2k 1 2k 2 where the sum is taken for t + t ≡ 1 (mod 2) (i.e. t and t are of opposite parity). Formula (53) is due to H. Wang, Z. Hu and L. Gao [474].

~~3.5 Cyclotomic polynomials~~

~~1 Introduction, irreducibility results~~

~~The nth cyclotomic polynomial n(x) is given by n(x) = (x − ζt )(1) t∈t(n)~~

2πit/n where ζt are the primitive roots of unity of order n, i.e. ζt = e for 1 ≤ t < n, (t, n) = 1. Another notation used in the literature, is Fn(x) for n(x), which has an old and long history. Since every nth root of unity is a primitive dth root of unity for some uniquely determined d|n, one has n x − 1 = d (x), (2) d|n which by the Mobius¨ inversion formula (see Chapter 2) implies n/d log n(x) = {log(x − 1)}µ(d), d|n giving n/d µ(d) n(x) = (−1 + x ) (3) d|n

> = a1 a2 ... ar If the prime factorization of n 1isn p1 p2 pr , then (3) implies / / ( n − )( n p1 p2 − )( n p1 p3 − )... ( ) = x 1 x 1 x 1 ( ) n x / / / 4 (xn p1 − 1)(xn p2 − 1)...(xn p1 p2 p3 − 1)... Formula (4) was proved by A. Cauchy in 1829 (see [103], p. 184), and it implies immediately that n(x) is of integer coefﬁcients (with leading coefﬁcient 1) with degree ϕ(n).

~~251 CHAPTER 3~~

~~From (3) (or (4)) follows that~~

~~p n(x ) np(x) = , if p n (p prime), (5) n(x)~~

~~p np(x) = n(x ), if p|n (6) For an odd prime p, (4) gives~~

~~x p − 1 (x) = = x p−1 + x p−2 + ···+x + 1,(7) p x − 1~~

(x2p − 1)(x − 1) (x) = = x p−1 − x p−2 + ···−x + 1, 2p (x p − 1)(x2 − 1) (x4p − 1)(x2 − 1) (x) = = x2p−2 − x2p−4 + ···−x2 + 1,(8) 4p (x2p − 1)(x4 − 1) etc., so one can write the following special cyclotomic polynomials: 1(x) = x − 1, 2 2 4 3 2 2(x) = x + 1, 3(x) = x + x + 1, 4(x) = x + 1, 5(x) = x + x + x + x + 1, 2 6 5 4 3 2 6(x) = x − x + 1, 7(x) = x + x + x + x + x + x + 1, 8(x) = x4 + 1, 6 3 4 3 2 10 9 8 7 6 9(x) = x +x +1, 10(x) = x −x +x −x+1, 11(x) = x +x +x +x +x + 5 4 3 2 4 2 8 7 5 4 3 x +x +x +x +x+1, 12(x) = x −x +1, 15(x) = x −x +x −x +x −x+1, 8 6 4 2 12 9 8 6 4 3 20(x) = x − x + x − x +1, 21(x) = x − x11 + x − x + x − x + x − x +1. For n > 1, n(0) = 1(i.e. constant term =1); p, if n = pa (p prime) (1) = (9) n 1, otherwise (i.e. ω(n) ≥ 2)

An important result is the irreducibility of the cyclotomic polynomial n(x) over the rational field. For n = prime, this was established by C. F. Gauss [146] in 1801. The first proof of the general case was given by L. Kronecker [232] in 1854. He proved actually a stronger result, namely that n is relatively prime to the discriminant of a polynomial with integer coefficients f (x), then n(x) remains irreducible over the field generated by any root of f (x).(10) There are many other irreducibility proofs for n(x).In1857, 1858, resp. 1859 R. Dedekind [100], F. Arndt [14], resp. V. A. Lebesgue [243]; for a survey of proofs up to 1907 see M. Ruthinger [358]. Further proofs are due to K. Grandjot (1924) [155], H. Spath¨ (1927) [435], E. Landau (1929) [238], I. Schur (1929) [401], F. Levi (1931) [254], T. Skolem (1949) [426], etc. L. Weisner [475] considered quadratic fields over which the cyclotomic polynomials are reducible, and more generally polynomials f (g(x)) reducible in fields in which f (x) is reducible [476]. The factorization of n(x) in quadratic fields and their composites was studied by D. Pumplun¨ [341],

252 THE MANY FACETS OF EULER’S TOTIENT who generalized previous work by H. Petersson [329]. Simple proofs of factorization (mod p) of n(x) may be found in R. Ballieu [25] or W. J. Guerrier [166]. In 1960 I. Seres [407] considered questions concerning the irreducibility of polynomials of the form n(P(x)), where P(x) is a polynomial. In [478] one can read that C. Nicol proved that p(x) + q (x) is irreducible for p, q primes; while generally for m, n > 1, if m(x) + n(x) factors, then for m, n ≤ 150, the factors contain a cyclotomic polynomial, e.g.

~~8 7 4 7(x) + 22(x) = 4(x)(x − x + 2x + 2)(11)~~

~~Is this generally true (or what conditions should be assumed)?~~

2Divisibility properties There are many divisibility properties of the cyclotomic polynomials or related objects. The following theorem is due essentially to L. Kronecker [233] (see also T. Nagell [306], p. 164-167): If p n is a prime, then

n(x) ≡ 0 (mod p)(12) is solvable iff p ≡ 1 (mod n).Ifp ≡ 1 (mod n), the solutions of congruence (12) are the numbers which belong to the exponent n(modulop).Ifx is a solution, then n the number n(x) is divisible by exactly the same power of p as x − 1. (13) The particular case n = pk was discussed by A. S. Bang in 1886 (see [103], p. 385). Bang proved also that if a > 1, n ≥ 3, then n(a) has a prime factor ≡ 1 (mod n) except for 6(2).(14) In 1905 L. E. Dickson (see [103], p. 388) showed that for a > 1, n(a) has a prime factor not dividing am − 1 (m < n) except in the cases n = 2, a = 2k − 1, and n = 6, a = 2. (15) The following result is due essentially to Dickson (see also [306] p. 166): Suppose a p is a prime factor of n, and put n = p n1, where p n1. Then the congruence (12) is solvable iff p ≡ 1 (mod n1).Inthis case, the solutions are the numbers which belong to the exponent n1(modulop).Ifx is a solution, then n(x) is divisible by p and not by p2, provided that n > 2. (16) More general results for the homogeneous form of n(x) were obtained by R. D. Carmichael in 1909. As a corollary of theorem (13) we note that: For n > 1 there are inﬁnitely many primes ≡ 1 (mod n).(17) Indeed, if there would be only a ﬁnite numbers of such primes, let P be their product. Put x = nPy(y > 0, integer). Then by (13), every prime factor of n(nPy) must

~~253 CHAPTER 3~~

be ≡ 1 (mod n). But this is impossible, since n(nPy) ≡ n(0) ≡ 1 (mod nP) (see [306], p. 168). Similar proofs of theorem (17) (i.e. a particular case of Dirichlet’s Theorem on primes in arithmetic progressions), by use of cyclotomic polynomials were obtained by A. S. Bang and J. J. Sylvester (see [103], p. 418). In 1896 A. Hurwitz (see [103], p. 378) proved the following primality criterion: If there exists an integer a such that

~~n−1(a) ≡ 0 (mod n)(18)~~

= = k + ( ) = 2k−1 + then a prime. For example, when n 2 1, since 2k x x 1, we get a theorem on the primality of Fermat numbers. We note that for n odd one has 2n(x) = n(−x),(19) and that for all n, 1 xϕ(n) = (x)(20) n x n which can be proved also by (3). (For an irrationality result based on (20), see [363], p. 286). In 1880 A. E. Pellet (see [103], p. 245) considered the equation

~~1 f (y) = 0,(y = x + , (x) = 0)(21) n x n i.e. the equation of degree ϕ(n) derived from n(x) = 0bythe substitution y = 1 x + .Heproved that if p n is a prime, and x~~

~~fn(y) ≡ 0 (mod p)(22)~~

2 has an integral root a, then n(x) is divisible modulo p by x − 2ax + 1. Either the latter has two real roots and n(x) and fn(y) have all their roots real and p − 1is divisible by n,oritisirreducible and n(x) is a product of quadratic factors (modulo p) and the roots of fn(y) are all real and p + 1isdivisible by n.Ifn divides neither p + 1 nor p − 1, fn(y) is a product of factors of equal degree (modulo p). (23) 1 J. J. Sylvester (1880) introduced similarly g (y) by setting y = x + in n x ϕ(n)/2 n(x)/(x ) (see [103], p. 384). He stated that every divisor of gn(y) is of the form ±1 (mod n), with the exception that, if n = pk(p ∓ 1)/m, then p is a divisor (but not p2). Conversely, every product of powers of primes of the form ±1 (mod n) is a divisor of gn(y). (24)

~~254 THE MANY FACETS OF EULER’S TOTIENT~~

In 1888 L. Kronecker and M. Bauer (see [103], p. 385) remarked that x + y (x − y)ϕ(n) = h (x, y2)(25) n x − y n is a polynomial with integer coefﬁcients, involving only even powers of y.Forthe prime factors of hn(x, s), the following is true (s is given): If p is prime, and p n, p s, then s p ≡ (mod n)(26) p In 1912 G. Fontene´ (see [103], p. 390) considered the homogeneous form a (a, b) derived from (x) by setting x = .Ifa and b are relatively primes, n n b then every prime divisor of n(a, b) is of the form ≡ 1 (mod n), unless it is divisible by the greatest prime factor p of n.Ithas this factor p if p−1isdivisible by n/pα α (where p n), and a, b satisfy n/pα (a, b) ≡ 0 (mod p), the latter having for each (b, p) = 1anumber of roots a equal to the degree of the congruence. In particular, a if n = p (p prime), every prime factor of n is of the form ≡ 1 (mod n), with the exception of a divisor p occurring if a ≡ b (mod p),and then to the ﬁrst power if n = 2. (27) When a, b are special complex numbers, results on the prime factors of n(a, b) have been obtained by T. N. Shorey and C. L. Stewart [413]. Let a, b ∈ C such that a (a +b)2 and a ·b are nonzero, relatively prime integers, with not a root of unity. Let b P(m) denote the greatest prime divisor of m,and let by convention P(0) = P(±1) = 1. Put Pn = P( n(a, b)) for n ≥ 3. Then for any k with 0 < k < 1/ log 2, and any integer n > 3, with at most k log log n distinct prime factors, we have

~~ω(n) Pn > C(ϕ(n) log n)/2 (28) where C > 0iseffectively computable in terms of a, b and k. For almost all integers n,~~

2 Pn > n(log n) /k(n) log log n,(29) where k : N → R is any function with the property k(n) →∞as n →∞. Finally we state certain simple divisibility properties related to n.Aswehave seen by relation (94 ) (see 5 of 3.2) one has n|ϕ( n(a)) for any a > 1. A similar property is due to A. Bartholome[28]:´

~~k k| n(a ) for inﬁnitely many n (30) for any (a, n) = (2, 1).~~

~~255 CHAPTER 3~~

~~R. Wang [473] proves that if p is a prime, and n > 2, n|p − 1; if p1,...,pϕ(n) are primes satisfying~~

n(pi ) ≡ 0 (mod p), i = 1,ϕ(n), (31) then ϕ(m) ≡ α or 1 (mod p) according as ω(n) = 1 and n is an αth power of a prime, or ω(n)>1, with m = p1 ...pϕ(n). E. Vantieghem [463] proves the congruence t (x − y ) ≡ n(x)(mod n(y)) for n ≥ 2 (32) t∈t(n) By using (32) with x = 1, y = 2heproves the following primality criterion: Let p > 2. Then p is prime iff

p−1 (2k − 1) ≡ p (mod (2p − 1)) (33) k=1 H. W. Leopoldt [252] rediscovered Bang’s result (14), and used it to solve a problem by Kostrikhin: ﬁnd all pairs (a, n) with a, n > 1 such that for all primes (34) p|(an − 1) one can ﬁnd an m < n such that p|(am − 1). For n = 2 all pairs (2s − 1, 2) with s ≥ 2 satisfy the property, while for n ≥ 3 the only pair is (2,6). (35) Cyclotomic polynomials are also used for factorization methods of integers, see e.g. E. Bach and J. Shallit [20], or R. P. Brent [52].

~~3 The coefﬁcients of cyclotomic polynomials We will now treat the coefﬁcients of cyclotomic polynomials. Let~~

~~ϕ(n) m n(x) = a(m, n)x (36) m=0 If n is prime, then a(m, n) = 1 for all m. When n = pq,with p, q distinct primes, then A. S. Bang [26] and A. Migotti [290] have proved that~~

~~a(m, n) ∈{−1, 0, +1} for all m (37)~~

Migotti showed also that a(7, 105) =−2. Since for all n ≤ 104, (37) holds true (see P. Erdos¨ [120]), 105(x) is the first cyclotomic polynomial which has other coefficients than 0 and ±1. The number m = 7isminimal, with |a(m, n)| > 1for some n,see M. Endo [108]. In fact, every integer appears as a coefficient of a certain cyclotomic polynomial, see J. Suzuki [450].

~~256 THE MANY FACETS OF EULER’S TOTIENT~~

Now, when n = pq, besides (37) there are known some special results. M. Beiter [34] has proved that (−1)δ, if n = αq + βp + δ in exactly one way, a(m, n) = (38) 0, otherwise where α, β are nonnegative integers and δ = 0, 1. L. Carlitz [62] proved that if p < q, the number of positive coefficients is equal to θ = (p − u)(uq + 1)/p (39) where u is defined by qu ≡−1 (mod p),0< u < p. Therefore, the total number of nonzero coefficients is

~~2θ − 1 = 2(p − u)(uq + 1)/p − 1 (40)~~

T. Y. Lam and K. H. Leung [237] have recently rewritten the results in another form. Write (p − 1)(q − 1) = rp+ sq (41) and let 0 ≤ k ≤ (p − 1)(q − 1). Then a(m, n) = 1iffk = ip + jq for some i ∈ [0, r]and j ∈ [0, s]; a(m, n) =−1iffk + pq = ip + jq for i ∈ [r + 1, q − 1] and j ∈ [s + 1, p − 1]; otherwise a(m, n) = 0. (42) The number of terms having ak = 1is(r + 1)(s + 1),and those with ak =−1is (p − s − 1)(q − r − 1). Furthermore, for p < q, the middle coefﬁcient is (−1)r .(43) Let A(n) = max{|a(m, n)| : m = 0, 1,...,ϕ(n)}. The following result is essentially due to Bang [26]: If n has three distinct odd prime factors p < q < r, then

~~An ≤ p − 1 (44)~~

If n has fewer than three odd prime factors, then An = 1. (45) D. M. Bloom [45] showed that if An = p − 1in(44), with p ≥ 5, then neither q nor r is ≡±1 (mod p).(46) Bloom proved also that if n has four odd prime factors p < q < r < s, then

~~An ≤ p(p − 1)(pq − 1)(47)~~

~~In 1931 I. Schur showed that (see [251])~~

~~lim sup An =+∞ (48) n→∞~~

~~257 CHAPTER 3~~

~~by proving that if m is any odd number and if n = p1 p2 ...pm, p1 < p2 < ···< pm, p1 + p2 > pm (pi primes), then~~

~~a(pm, n) = m − 1 (49)~~

E. Lehmer [251] showed that (An) is unbounded, even when n is a product of only three primes, by proving that if p, q = kp + 2andr = (mpq − 1)/2areall primes, and if h = (p − 3)(qr + 1)/2, then

p − 1 a(h, pqr) = (50) 2 W. Bosma [47] has recently used Magma system for computational algebra to list for |a|≤50 the least n for which a appears as a coefficient of n(x).Infact, values n ≤ 26565 suffice for all a with |a|≤50 with the exception of a =−50, which 1082 occurs for the first time in 40755(x) as coefficient of x . ϕ(n) ϕ( ) = ,ϕ( ) α = α ( ) = k n has n roots x j , j 1 n .Let k k n x j be the Ramanujan j=1 sum (denotes also as C(k, n)). By using Newton’s formulae concerning symmetric polynomials, we can write

~~α1b0 + b1 = 0 α2b0 + α1b1 + 2b2 = 0 ... (51) αi b0 + αi−1b1 + ···+ibi = 0 ...~~

~~αϕ(n)b0 + αϕ(n)−1b1 +···+ϕ(n)bϕ(n) = 0 where~~

~~ϕ(n) m ϕ(n) ϕ(n)−1 n(x) = a(m, n)x = b0x + b1x +···+bϕ(n) = 0 (52) m=0~~

We may view (51) and (52) as a system of equations in the ϕ(n) + 1 unknowns b0, b1,...,bϕ(n) (x being an arbitrary primitive root of unity), and since b0 = 1, the determinant of thus homogeneous system must be zero. By developing this determinant under the ϕ(n) + 1throw (i.e. the one which contains xϕ(n), xϕ(n)−1,...,1) one gets that

~~(− ) (− )i (− )ϕ(n) ϕ(n) 1 ϕ(n)−1 1 ϕ(n)−i 1 (x) = x + A x +···+ A x +···+ Aϕ( ) (53) n 1! 1 i! i n! n~~

~~258 THE MANY FACETS OF EULER’S TOTIENT~~

where Ai (obtained by Laplace’s rule) is given by α ... 1 1000 0 α α ... 2 1 200 0 α3 α2 α1 30... 0 Ai = (i = 1,ϕ(n)) (54) ...... α − α − α − ...... i − 1 i 1 i 2 i 3 αi αi−1 αi−2 ...... α1

Since by Holder’s¨ theorem n n α (n) = ϕ(n)µ /ϕ (55) k (n, k) (n, k) by (54), all coefficients of n(x) can be determined (at least, in theory), see M. Dea- conescu and J. Sandor´ [98] or D. H. Lehmer [246]. For example, in (52) one has  ( − ),  u u 1 if n is odd  u(u − 1) =−µ( ), = , if n = 2m, m odd ( ) b1 n b2  2 56  u − , if n = 2km, m odd, k > 1 2 where u = µ(m). A complicated formula can be written for b3. When (n, 6) = 1, 1 b =− (u3 − 3u2 + 2u)(57) 3 6 for a particular table of values up to b10, see [246]. By using Bernoulli and Stirling numbers,Lehmer determines exactly also the (n) coefficients of n(x + 1).For example, the constant term is e ; the coefficient 1 of x is ϕ(n)e(n), where (n) is the von Mangoldt function. (These were known 2 essentially to Lebesgue and Holder).¨ The coefficient of xk is

(n) e Rk{ϕ(n), J2(n), J4(n), ...,J2h(n)} (58) where 2h ≤ k < 2h + 2, Rk is a polynomial with rational coefﬁcients, and Jm(n) is Jordan’s totient function. k Bounds for the coefﬁcients a(k, n) of x in n(x) given by (36) were obtained by P. Erdos¨ and R. C. Vaughan [131]:

~~1/2 3/8 |a(k, n)| < exp{ck + c1k } (59)~~

~~259 CHAPTER 3 and / 1 k 1 2 lim sup |a(k, n)| > exp c2 , k > k0,(60) n→∞ 625 log k~~

~~/ 2 1 2 where c = 2 1 − , and c , c are other constants. 2 + 1 2 pprime p p When n(x) is given by (52), H. Moller¨ [293] proved that~~

− 2s 4 s−2 2q s−k−1 |b (q q ...q )|≤ 1 · (q − )2 −1,(s ≥ )() i 1 2 s − k 1 4 61 q1 1 k=1 where q1 < ···< qs are arbitrary odd primes. Schur’s theorem (48) was improved for the ﬁrst time in 1946 by P. Erdos¨ [120]:

~~4/3 An > exp{c1(log n) } for inﬁnitely many n (c1 > 0)(62)~~

~~Later [121] he reﬁned this to c log n A > exp exp for inﬁnitely many n (c > 0)(63) n log log n~~

P. T. Bateman [32] proved that log n A < exp exp(log 2 + o(1)) (64) n log log n and R. C. Vaughan [466] showed that the constant log 2 is best possible. Similarly c = log 2 in (63) is best possible. (65) A(n) depends essentially on the squarefree kernel of n.Bateman, Pomerance and Vaughan [33] have showed that if s ≥ 3, and q1 < ···< qs are odd primes, then

~~s−2 S(q ...q ) s−k−1 1 s < A(q ...q )< q2 −1 (66) ... 1 s k q1 qs k=1~~

~~ϕ(n) where S(n) = |a(m, n)|. m=0 In 1990 H. Maier [274] stated the following result: Let ε(n) be a function deﬁned for all positive integers n such that ε(n) → 0asn →∞. Then~~

~~A(n) ≥ nε(n) for almost all n (67)~~

~~260 THE MANY FACETS OF EULER’S TOTIENT~~

~~In another paper [275] it is shown that for any N > 0 there exist numbers c(N)> 0, x0(N) ≥ 1 such that~~

~~card{n ≤ x : A(n) ≥ n N }≥c(N)x (68)~~

~~In [276] it is considered the following counterpart of (67): Let ψ(n) be a function deﬁned for all positive integers n such that ψ(n) →∞for n →∞. Then~~

~~A(n) ≤ nψ(n) for almost all n.~~

~~Some recent work has been concerned with~~

~~B(m) = max{|a(m, n)| : n ≥ 1}=lim sup |a(m, n)|. n→∞ In 1985 H. L. Montgomery and R. C. Vaughan [296] proved that~~

log B(m) m1/2(log 2m)−1/4 (69) log B(m) m1/2(log m)−1/4 √ which show that log B(m) has order of magnitude m(log m)−1/4 for large m. G. Bachman [21], improving (69) showed that √ m log log m log B(m) = C 1 + O (70) (log m)1/4 log m where C is an explicitly given constant. The proof includes the Hardy-Littlewood circle method and estimates on exponential sums with multiplicative coefﬁcients. Finally we note that L. Carlitz [63] studied questions related to the sum of squares of n(x).

~~4 Miscellaneous results This last section includes miscellaneous results on cyclotomic polynomials. Certain inequalities for n(x) are:~~

~~ϕ(n) ϕ(n) (x − 1) < n(x)<(x + 1) for x > 1, n > 1 (71)~~

~~(see [252]) and~~

~~r−2 r−2 r−1 x − 1 2 x2 − 1 2 x 2 xϕ(n) < (x)< xϕ(n), x > 1, n > 1, x x2 n x − 1 (72) where r = ω(n) (see R. Richter [351]).~~

~~261 CHAPTER 3~~

~~If n ≡ 1 (mod 2), then~~

~~r−1 r−2 r−2 x 2 x + 1 2 x2 + 1 2 xϕ(n) < (−x)< xϕ(n); (73) x + 1 n x x2 if n ≡ 2 (mod 4), n > 2, then~~

r−3 r−3 r−2 x − 1 2 x2 − 1 2 x 2 xϕ(n) < (−x)< xϕ(n) (74) x x2 n x − 1 If n is an odd multiple of a prime ≡ 1 (mod 4), then for sufﬁciently large n (see [247]): n(−2)>n(2n + 1)(75)

Consider the function f :[1, ∞) → R, f (t) = n(t), where n is the nth cyclotomic polynomial. K. Motose [304] proved that f is a strictly increasing function for all n.(76) A theorem of T. M. Apostol [11] and F.-E. Diederichsen [105] gives the value of the resultant of cyclotomic polynomials m and n for m > n > 1asfollows: m pϕ(n), if n|m and is a power of a prime p, ρ( m(x), n(x)) = n (77) 1, otherwise

Foranew proof, see S. Louboutin [260]. More generally, Apostol [12] showed that for all m, n ≥ 1and arbitrary nonzero complex numbers a, b ϕ(m)ϕ(n) d d µ( n )ϕ(m)/ϕ(m/δ) ρ( m(ax), n(bx)) = b [ m/δ(a /b )] d (78) d|n where δ = (m, d).When m and n are distinct primes p and q, then (78) simpliﬁes to   a pq − b pq a − b · , for a = b ρ( ( ), ( )) = p − p q − q ( ) q ax p bx  a b a b 79 a(p−q)(q−1), for a = b

M. Kaminski [218] proved that if P(x) = x is a monic irreducible polynomial with integer coefﬁcients such that ρ(P(x), n(x)) =±1 for inﬁnitely many n, then P(x) is a cyclotomic polynomial. A connection between cyclotomic polynomials and the Riemann hypothesis has been studied by F. Amoroso [9], [10]. Let FN (z) = n(z) and put n≤N π 1 + iθ h(FN ) = log |FN (e )|dθ(80) 2π −π

~~262 THE MANY FACETS OF EULER’S TOTIENT~~

(a function introduced by K. Mahler in 1964 and M. Mignotte in 1979 in the Theory of Diophantine approximations). Amoroso shows that the inequality λ+ε h(FN ) N (ε > 0)(81) is equivalent to the assertion that the Riemann zeta function does not vanish for Re z ≥ λ + ε.(82) 1 (Particularly, the Riemann hypothesis states (82) for λ = ). 2 λ< | (α)|− λ Another result in the same vein is that if 1 and if log FN N for every root of unity α of order ≤ N, then for any ε>0, the Riemann zeta function does not vanish for Re z ≥ λ + ε.(83) V. N. Sorokin [434] proved the linear independence over Q of the set {log d (a, b) : d|n}∪{1}, where n is a positive integer; a, b are two nonzero rational n+1 n 1/(p−1) integers satisfying |b| > cn|a| , where cn = 12(2nεn) , with εn = p (p p|n prime). (84) R. Creutzburg and M. Tasche [90] proved that a Gaussian integer e is a primitive nth root of unity modulo a Gaussian integer m iff m is a divisor of n(e)/q, where q is either 1, p or π, where p is the largest prime divisor of n, and π is a Gaussian prime divisor of p.(85) Let α be a unit of degree d in an algebraic number ﬁeld and assume that α is not a root of unity. Let U(α) be the number of values of n for which n(α) is a unit. Then J. H. Silverman [417] proved that

~~U(α) ≤ cd1+0.7/ log log d ,(86) where c is an effectively computable constant.~~

~~3.6 Matrices and determinants connected with ϕ~~

~~1 Smith’s determinant In 1876 H. J. S. Smith [432] discovered his famous determinant theorem~~

~~det[(i, j)]n×n = ϕ(1)ϕ(2)...ϕ(n)(1) where (i, j) is the greatest common divisor (GCD) of i and j. In fact (see [432] and [103], p. 128) Smith proved the more general result~~

~~det[ f (i, j)]n×n = g(1)g(2)...g(n), where f (m) = g(d), m = 1, 2,... (2) d|m~~

~~263 CHAPTER 3~~

(and where we have denoted by f (i, j) the value taken by f at (i, j), i.e. f ((i, j))). When f (m) = m,byGauss’ identity (1) follows. It should be noted that relation (2), or particular cases of it, has been rediscovered many times in the literature. Let f (n) = nk (k ∈ R). Then mk = g(d) givesbyMobius¨ inversion g(n) = | d m m k µ(d) 1 µ(d) = mk = mk − = J (m) k 1 k k , i.e. the extension of d|n d d|n d p|n p Jordan’s totient for k ∈ R. One obtains k det[(i, j) ]n×n = Jk(1)Jk(2)...Jk(n), (3) rediscovered by another method by B. Gyires in 1957 [172]. In 1965 I. Gy. Mau- rer and M. Veegh´ [286] gave two new proofs of (3); one is based on an inductive argument; the other proof is in fact the classical one. Forasurvey of generalizations of Smith’s determinant up to 1961, see Gy. Olah´ [325]. Let f (n) = d(n), the number of divisors of n. Then

det[d(i, j)]n×n = 1,(4) which is included in Polya-Szeg´ o¨ (1924 the ﬁrst edition) Part VIII, Problem 1.31. Problem 1.33 is in fact (2), with a proof different from Smith’s. The proof is based on a remark that [ f (i, j)]n×n can be written in the form

~~T [ f (i, j)]n×n = B · C (5) where B and C are lower triangular matrices given by bij = g( j) if j|i;and0, otherwise; cij = 1if j|i; and 0, otherwise. L. Carlitz [64] observed that~~

T [ f (i, j)]n×n = (diag(g(1), g(2), . ..,g(n))C (6) where C is the triangular matrix from (5). Relations (3), (4) appear also in [323], where one can ﬁnd also some other particular cases. Put e.g. f (n) = σ(n), the sum of divisors of n. Then, since g(d) = d, one has det[σ(i, j)]n×n = n! (7) Let f (n) = µ(n), the Mobius¨ function. Then it is immediate that g(p) =−2, g(p2) = 1 and g(pr ) = 0 for r ≥ 3(p prime), and g is multiplicative, so one has

~~det[µ(i, j)]n×n = k(n), (8) where k(n) = 1 for n = 1; −2 for n = 2;4forn = 3; 4 for n = 4; −8 for n = 5; −32 for n = 6; 64 for n = 7; and 0 for n ≥ 8. (9)~~

~~264 THE MANY FACETS OF EULER’S TOTIENT~~

Let f (n) = γ(n) = n, the core of n. Then g(p) = p − 1, g(pr ) = 0for p|n r > 1, so: det[γ(i, j)]n×n = l(n), (9) where l(n) = 1, for n = 1;1forn = 2;3forn = 3; and 0 for n ≥ 4. 1 1 [i, j] Let f (n) = . Since = ,where [i, j] denotes the l.c.m. of i, j;by n (i, j) ij 1 g(n) = f (n) − using properties of determinants, and the fact that 1 ( ) , p|n f p when f is multiplicative, one can deduce: det [i, j] = ϕ(1)ϕ(2)...ϕ(n)γ (1)...γ(n)(−1)ω(1)+···+ω(n) (10) n×n where γ(k) is the core of k,andω(k) is the number of distinct prime factors of k.We note that (10) is due also to Smith [432], who used the notation γ(k)(−1)ω(k) = π(k) (i.e. it is a multiplicative function satisfying π(1) = 1, π(pr ) =−p for r ≥ 1). Example (1), (4), (7), as well as a proof of (2) can be found also in E. D. Schwab [402]. The determinant of (10) has been presented also in Problem 10232 of Ameri- can Math. Monthly 96(1992), no. 6, as a subject for evaluation. Problem 6089 by R. M. Redheffer [347] however shows that a small change in the determinant from (4) can lead to a serious difference. Namely, 2 det[d(i, j)]2≤i, j≤n = µ (n), (11) i≤n i.e. if one deletes the ﬁrst line and ﬁrst row in (4), then one obtains that the value of the obtained determinant is equal to the number of squarefree integers from 1 to n. In 1972 T. M. Apostol [13] by using the idea that [ f (i, j)]n×n can be written as a product of two triangular matrices, generalized (2) as follows: Let f, g and h satisfy f (i, j) = g(d)h( j/d)(12) d|(i, j) Then n det[ f (i, j)]n×n = g(1)g(2)...g(n)(h(1)) (13) By taking g(n) = n (n = 1,...) and h = µ, one obtains

~~det[C(i, j)]n×n = n! (14) where C(i, j) is Ramanujan’s sum (denoted by C(n, d) in (27) of 3.1). A more general Ramanujan sum is k Ck(n, d) = e(nx, d ), (15) k (x,d )k =1~~

~~265 CHAPTER 3~~

~~where the sum is over all x in an arbitrary reduced (d, k)-residue system, and (a, b)k is the largest kth power divisor of a and b.Then (see McCarthy [69])~~

det(Ck(i, j)]n×n = 0 (16) for all n, k ≥ 2. For more general results, where Apostol’s evaluation is extended to the case of even arithmetical functions, see P. J. McCarthy [71]. G. Daniloff [93] found an analogue of Smith’s determinant as follows: Let k Dk(n) = m if n = m for some positive integer m, and =0 otherwise. Let f (i, j) = Dk(i/d)Dk( j/d)g(d)(17) d|(i, j) Then det[ f (i, j)]n×n = g(1)g(2)...g(n)(18) The unitary analogue of Smith’s determinant was introduced by H. Jager [211]. ∗ Let (i, j) denote the greatest common unitary divisor of i and j. Let d m denote m that d is a unitary divisor of m (i.e. d|m, d, = 1; see Chapter 2). If d f (i, j) = g(d), (19) d (i, j)∗ then det[ f (i, j)] × = g(1)g(2)...g(n)(20) n n Let ϕ∗ be the unitary totient function. Since ϕ∗(d) = m,Jager deduced from d m (20) the analogue of (1):

~~det[(i, j)∗] = ϕ∗(1)ϕ∗(2)...ϕ∗(n)(21)~~

K. Nageswara Rao [308] gave an A-analogue of Smith’s determinant, where A denotes Narkiewicz’s regular convolution. See also C. R. Wall [471]. The classical and the unitary cases are particular cases of this A-analogue determinant. Multidimensional Smith’s determinants have been considered by L. Gegenbauer [148], D. H. Lehmer [248], N. P. Sokolov [433], P. Haukkanen [181], R. Vaidyan- thaswamy [462].

~~2Poset-theoretic generalizations Smith deduced his result (2) in a slightly more general setting, namely if x1, x2,...,xn are distinct positive integers with the property that~~

~~d|xi ⇒ d = x j for some j = 1, 2,...,n,(22)~~

~~266 THE MANY FACETS OF EULER’S TOTIENT then~~

~~det[ f (xi , x j )]n×n = g(x1)g(x2)...g(xn)(23)~~

~~Such a set S ={x1, x2,...,xn} (i.e. satisfying (22)) is called factor-closed set. Forexample,~~

det[(xi , x j )]n×n = ϕ(x1)ϕ(x2)...ϕ(xn)(24) for a factor closed set S. The idea has been extended to the lattice-theoretical context by H. S. Wilf [482], B. Lindstrom¨ [259], P. Haukkanen [182], B. V. Rajarama Bhat [342], P. Haukkanen [183], P. Haukkanen, J. Wang and J. Sillanpaä¨ [192]. Forexample, a common generalization of (13), (20) and (23) is obtained in the context of meet semilattices. Let (P, ≤) be a poset (see Chapter 2). Then P is called a meet semilattice if for any x, y ∈ P there exists a unique z ∈ P such that: 1) z ≤ x and z ≤ y;2)ifw ≤ x and w ≤ y, for some w ∈ P, then w ≤ z. In such a case z is called the meet of x and y, and denoted by x ∧ y.LetS ⊂ P. The set S is called lower-closed if for every x, y ∈ P with x ∈ S, y ≤ x,wehave y ∈ S.Wenote that in the case of (N, |), the lower closed sets coincide with the factor- closed sets of (22). The following result is true: Let P be a finite meet semilattice, and let {x1, x2,...,xn} be a lower-closed subset of P.LetF, G : P × P → C be incidence functions of P, and let A be the n × n matrix defined by aij = F(x, xi )G(x, x j ). (25) x≤xi ∧x j

~~Then n det A = F(xk, xk)G(xk, xk)(26) k=1~~

Let eD(d, m) = 1isd|m;=0otherwise in the poset (N, |). Letting F(d, m) = eD(d, m), G(d, m) = eD(d, m)g(d). Then f (xi , x j ) = n F(d, xi )G(d, x j ) = F(xk, xi )G(xk, x j ), and the above theorem yields d|(xi ,x j ) k=1 (23). Let eu(d, i) = 1ifd i;=0otherwise in the poset (N, ).Thenasabove,we reobtain from (26) relation Jager’s theorem (20). If F and G are incidence functions of (N, |),letF(d, i) = eD(d, i)g(d); G(d, j) = eD(d, j)h( j/d).Then we reobtain Apostol’s theorem (13). Let C∗(i, j) = dµ∗( j/d) be the unitary analogue of Ramanujan’s d|i,d j sum, where µ∗ is the unitary Mobius¨ function (see Chapter 2). Denoting ∗ ∗ F(d, i) = eD(d, i)d and G(d, j) = eu(d, j)µ ( j/d).Bywriting C (i, j) =

~~267 CHAPTER 3~~

~~n F(k, i)G(k, j),itfollows that k=1 ∗ det[C (i, j)]n×n = n! (27)~~

Foralast application, let eN (k, i) = 1ifk ≤ i;=0otherwise on the poset (N, ≤). Then n min{i, j}= eN(k, i)eN(k, j) = eN(k, i)eN(k, j), k≤min{i, j} k=1 so: det[min{i, j}]n×n = 1 (28) The subset S ⊂ P of a finite meet semillatice will be called meet-closed (see [192]) if for every x, y ∈ S we have x ∧ y ∈ S. Let f : P → C.Thenthe matrix, defined by (S) f = (sij) with sij = f (xi ∧ x j )(1 ≤ i, j ≤ n) is called the meet-matrix of S with respect to f . The following theorem is due to Rajarama Bhat [342]: Let S ={x1,...,xn}⊂P be a meet-closed subset, and let g(xi ) be defined by g(xi ) = f (xi ) − g(x j ), (29) x j ∈S,x j

(x j < xi means x j ≤ xi and x j = xi ), where f is given above. Then det(S) f = g(x1)...g(xn)(30) The following corollary of (30) contains earlier results by Wilf [482] and Lind- strom¨ [259]: Let S be lower-closed subset of P. Then

det(S) f = g(x1)...g(xn), (31) where g(xi ) = f (x j )µ(x j , xi ), x j ≤xi µ being the Mobius¨ function of P. This is a poset-theoretic generalization of Smith’s theorem. The incidence matrix of two subsets S ={x1,...,xn} and T ={y1,...,ym} is deﬁned by E(S, T ) as an n × m matrix whose i, j entry is 1 if y j ≤ xi ; and 0, otherwise. Let S ={x1,...,xn, xn+1,...,xn+r } be the unique (up to isomorphism) minimal meet semilattice containing S.Ifg : S → C is deﬁned as in (29), then

~~T (S) f = Ediag(g(x1), g(x2), . ..,g(xn+r ))E (32)~~

268 THE MANY FACETS OF EULER’S TOTIENT where E = E(S, S) and E T is the transpose of E. This result is from [342], and is a generalization of Carlitz’s theorem (6). Theorems (30) and (32) extend results of S. Beslin and S. Ligh [41], [42], where the classical (N, |) poset is considered. In [192] the determinant of (32) is calculated:

~~∗ ( ) = ( )2 ( )... ( )() det S f det E(k1,...,kn) g xk1 g xkn 33~~

∗ ≤ < < ··· < ≤ + where in one has 1 k1 k2 kn n r,andE(k1,...,kn) is the submatrix of E = E(S, S) consisting of the k1th, ... ,knth columns of E. If f and g are real valued functions and g(x) ≥ 0, then

det(S) f ≥ g(x1)...g(xn)(34) with equality iff S is a meet-closed subset. Let P be a ﬁnite meet semillatice, S ={x1,...,xn}⊂S, and f : P → C. The following generalized totient function has been introduced in [342]. Let ϕS, f be deﬁned inductively by ϕS, f (x j ) = f (x j ) − ϕS, f (xi )(35) xi

~~269 CHAPTER 3~~

3Factor-closed, gcd-closed, lcm-closed sets, and related determinants We now return to the classical poset (N, |), where further analogous notions and results are obtainable. Let S{x1,...,xn} be a set of distinct positive integers. Let the G ⊂ D matrix whose i, j-entry is (xi , x j ) be denoted simply by (S),andin the general case of f (xi , x j ),by(S) f ( f being an arithmetic function). Similarly, let the LCM matrix whose i, j-entry is [xi , x j ]bedenoted by [S], and in the case of f [xi , x j ], by [S] f . As we have seen in section 2, S is said to be factor-closed if d ∈ S whenever xi ∈ S and d|xi .H.J.S.Smith showed that if S is factor-closed, then (2) holds. S. Beslin and S. Ligh [41] have called S gcd-closed if (xi , x j ) ∈ S, whenever xi , x j ∈ S. Note that if S is factor-closed, then it is gcd-closed. too. They extended (2) to the case of gcd-closed sets (in fact, (30) of section 2 for meet-matrices, generalizes this result): If S is gcd-closed set, then

n det(S) f = ( f ∗ µ)(d)(38) i=1 d where d runs through the divisors of xi such that d xt for xt < xi .Wenote that Beslin and Ligh obtained (38) for the case f (n) = n, this general form is due to Bourque and Ligh [48]. By extending (10) of Smith to gcd-closed matrices, they proved also that (see [49])

n = 2 ϕ( )γ ( )(− )ω(d)/ 2 ( ) det[S] xi d d 1 d 39 i=1 d where the sum over d is as in (38). For determinants of matrices on gcd-closed sets, and associated to other arithmetic functions, see S. Hong [202]. See also [203] by Hong on LCM matrices. When S is factor-closed, and f is a multiplicative function, in another paper [50] they showed that

n 2 det[S] f = f (xi ) g(xi ), (40) i=1 1 where g = ∗ µ, and f (n) = 0 for all n. f For f (n) = nk this gives a result (similar to (10)) due also to H. J. S. Smith. In 1996 P. Haukkanen and J. Sillanpa¨a¨ [187] extended (40) to gcd-closed sets S, but for quasi-multiplicative functions, deﬁned as follows: f is said to be quasi- multiplicative, if f is not identically zero, and if there exists a non-zero constant q such that f (a) f (b) = qf(ab)(41)

270 THE MANY FACETS OF EULER’S TOTIENT whenever (a, b) = 1. When q = 1 this gives the usual notion of multiplicative functions. Now let S be a gcd-closed set, and suppose that f is quasi-multiplicative, and f (n) = 0 for all n. Then

n 2 det[S] f = f (xi ) g(d), (42) i=1 d where g is given as in (40), and the sum over d is as in (38). Haukkanen and Sillanpa¨a¨ introduced also the concept of lcm-closed sets S, and obtained results for det(S) f and det[S] f when f is completely multiplicative. S is said to be lcm-closed if [xi , x j ] ∈ S for every xi , x j ∈ S.IfS is lcm-closed, then xi | max S for all xi ∈ S. Let us denote max S = xn = m.Now the following theorems are valid: Let S ={x1,...,xn} be an lcm-closed set of distinct positive integers, and let f be completely multiplicative with f (n) = 0 for all n. Then

~~n −n 2 det(S) f = f (m) f (xi ) ( f ∗ µ)(d)(43) i=1 d m m where d runs through the divisors of such that d for xi < xt . xi xt 1 Let S and f be as above, and let g = ∗ µ. Then f~~

n n det[S] f = f (m) g(d)(44) i=1 d where in the sum d is as in (43). Regarding unitary Smith determinants, we have stated results like (19) and (27). In what follows, (a, b)∗ will denote the greatest common unitary divisor (gcud) of a and b, while (a, b)∗ will denote the semi-unitary greatest common divisor (sugcd), which is the greatest unitary divisor of b which is a divisor of a. The least common unitary multiple (lcum) of a and b will be deﬁned by [a, b]∗ = ab/(a, b)∗. The set S ={x1,...,xn} of distinct positive integers will be called ud-closed if ∗ d ∈ S whenever d xi for xi ∈ S. The set S is said to be gcud-closed if (xi , x j ) ∈ S for every xi , x j ∈ S; and sugcd-closed if (xi , x j )∗ ∈ S for every xi , x j ∈ S. Haukkanen and Sillanpa¨a[¨ 187] prove the following theorems: If S is gcud-closed, and f is an arbitrary arithmetic function, then

~~n ∗ ∗ det(S) f [ f ((xi , x j ) )]n×n = ( f ⊕ µ )(d)(45) i=1~~

271 CHAPTER 3 where ⊕ is the unitary convolution, µ∗ the unitary Mobius¨ function, and the sum is over all d xi , d ∦ xt for xt < xi . As a corollary, (45) implies that if S is ud-closed, then n ∗ ∗ det[ f ((xi , x j ) )]n×n = ( f ⊕ µ )(xi )(46) i=1 containing the particular case n ∗ ∗ det[(xi , x j ) ]n×n = ϕ (xi )(47) i=1 For the next theorem we must assume that f is completely multiplicative. Let S be gcud-closed, and let f be completely multiplicative with f (n) = n for all n. Then n ∗ 2 1 ∗ det[ f ([xi , x j ] )]n×n = f (xi ) ⊕ µ (d), (48) i=1 f where the sum is over d as in (45). Result (45) holds true also if S is sugcd-closed set. The corresponding result to (48) for sugcd-closed sets, however, is an open problem. For the singularity or nonsingularity of matrices related to (S) and [S], see [49], [188], [201], [192]. The reciprocal GCD and LCM matrices have been introduced by S. Beslin [40]. These are the matrices 1 , 1 ( ) ( , ) resp. , 49 xi x j n×n [xi x j ] n×n Forexample, when S is factor closed, then 1 n det = x−2ϕ(x )(50) , i i [xi x j ] n×n i=1 r For more general results, with [xi , x j ] in (50), see E. Altinis¸ik and D. Tas¸ci [6]. For determinants of type f (i, j) det (51) ij i|n, j|n see P. Codeca´ and M. Nair [75]. In [231] A. Krieg studies the number N(x) of matrices M in the modular group SL2(Z) whose maximum norm ≤ x.Itisshown that N(x) is basically the partial sum of Euler’s totient function, therefore 96 N(x) ∼ x2 (x →∞)(52) π 2

~~272 THE MANY FACETS OF EULER’S TOTIENT~~

~~4 Inequalities~~

Relation (34) gives an inequality for det(S) f , when S is a subset of a meet semilattice. In [48] Bourque and Ligh proved that if S ={x1,...,xn} is a set of distinct positive integers, and if ( f ∗ µ)(d)>0forall d ∈{d : d|x, x ∈ S}, ( f : N → R), then n det[ f (xi , x j )] ≥ ( f ∗ µ)(xk)(53) k=1 with equality only if S is factor-closed. This result was first proved by Z. Li [255] in the case f (x) = x for all x. P. Haukkanen [183] obtained an extension to the case of meet-closed sets (see section 2): Let T ={y1,...,ym} be a meet-closed set containing S ={x1,...,xn} and let f (w)µ(w, z)>0forall z ∈{z : z ≤ y, y ∈ T }. Then w≤z n det(S) f ≥ ϕT, f (xk), (54) k=1 with equality iff S is meet-closed, and ∀ x ∈ S: ∀ z ≤ x : z ∈ T \ S (see (35) for the definition of ϕT, f ). Letting T = S in (54) one obtains: If S ={x1,...,xn} and f (w)µ(w, z)>0 for all z ∈{z : z ≤ y, y ∈ S}, then w≤z n ( ) ≥ ϕ ( )() det S f S, f xk 55 k=1 This gives in the case of the poset (N, |) an improvement of (53): n det[ f (xi , x j )] ≥ ( f ∗ µ)(d)(56) k=1 where the sum is taken over all d|xk, d yt for yt < xk; and all conditions are as in theorem (53). Z. Li [255] proved also the upper bound n! det[(x , x )] ≤ x x ...x − (57) i j 1 2 n 2 In [48] it is proved that for GCD matrices associated with arithmetical functions satisfying the condition ( f ∗ µ)(d)>0forall d ∈{d : d|x, x ∈ S},wehave that (S) f is positive definite, and thus

~~det[ f (xi , x j )] ≤ f (x1)... f (xn)(58)~~

~~273 CHAPTER 3~~

In [183] it is shown that this holds also for the more general case of meet matrices of poset-theoretic setting: If S ⊂ P and f (w)µ(w, z)>0 for all z ∈{z : z ≤ w≤z x, x ∈ S}, then (S) f is positive deﬁnite, so

~~det(S) f ≤ f (x1)... f (xn)(59)~~

By [50] similar bounds are also true for det[ f [xi , x j ]]: Let f : N → R be multiplicative and assume that (g ∗ µ)(d)>0forall d ∈{d : d|x, x ∈ S}, with 1 g = . Then [ f [x , x ]] is positive deﬁnite, and f i j n n 2 1 f (xk) ∗ µ (xk) ≤ det[ f [xi , x j ]] ≤ f (xk), (60) k=1 f k=1 with equality on the left side iff S is factor closed. In [6] it is proved that n J (x ) 1 n 1 r k ≤ det ≤ (61) 2r , r r k=1 xk [xi x j ] k=1 xk where r > 0, and Jr is Jordan’s totient function. The reciprocal GCD and LCM matrices were introduced in [40]. These are de- 1 1 noted by (S−1) = , resp. [S−1] = . See also [49] and [230]. (xi , x j ) [xi , x j ] Let · 2 denote the Euclidean norm of a matrix. D. Bozkurt and S. Solak [51] have shown that ! 5π 2 [S−1] ≤ − 4 (62) 2 6 Recently, E. Altinisik, N. Tuglu and P. Haukkanen [7] have generalized and improved (62) to ζ(rp)3/p [S−r ] ≤ ,(rp > 1), (63) p ζ(2rp)1/p ζ · ( ≥ ) where is the Riemann zeta function, and p is the l p norm p 1 of matrices 1/p n 1 (if A = [a ], then A = |a |p ). Here [S−r ] = .Forr = 1, ij p ij [x , x ]r i, j=1 √ i j 15 p = 2, the right side of (63) gives π, improving (62). 6 The spectral norm of a matrix A = [aij]isdeﬁnedby √ A ∗ = max{ λ : λ is an eigenvalue of A∗ A}.

~~274 THE MANY FACETS OF EULER’S TOTIENT~~

~~Let 2r > 1. Then (see [7])~~

~~ζ(2r)3/2 [S−r ] ∗ ≤ (64) ζ(4r)1/2 For inequalities on GCUD-reciprocal LCUM determinants, see the recent paper by A. Nalli [311].~~

~~3.7 Generalizations and extensions of Euler’s totient~~

1Jordan, Jordan-Nagell, von Sterneck, Cohen-totients There exist many generalizations, extensions, or analogues functions to Euler’s totient. In Chapters 1 and 2, as well as in the former paragraphs of this Chapter 3, there have been included many notions or results related to totients. Perhaps the most important generalization of Euler’s totient is the Jordan totient function Jk with Jk(n) deﬁned as the number of ordered sets of k elements from a complete residue system (mod n) such that the g.c.d. of each set is prime to n.Infact

Jk = µ ∗ Ek (see 2.2.1), so 1 J (n) = nk − ( ) k 1 k 1 p|n p ForJordan’s totient, see 2.2.1, 2.2.7 of Chapter 2, and 3.1.1, 3.1.7, 3.2.11, 3.3.5, 3.5.3, 3.6.1 of Chapter 3. An extension of Jordan’s totient, namely the Jordan-Nagell totient occurred at 2.3.3 of Chapter 2. Two generalized totients strongly related to Jordan’s are the von Sterneck and the Cohen totients.In1894 R. D. von Sterneck (see [103], p. 151) deﬁned Hk(n) = ϕ(d1)ϕ(d2)...ϕ(dk)(2) n=[d1,d2,...,dk ] where the summation is over all sets of k positive integers d1,...,dk with their lcm equal to n.Heproved that Hk(n) = Jk(n)(3) In 1952 E. Cohen [78] introduced the notion of a k-reduced residue system (mod n). Let a, b be integers, not both zero. (a, b)k denotes the g.c.d. of a and b which is also a kth power. If (a, b)k = 1, a is said to be relatively k-prime to b

275 CHAPTER 3 and vice-versa. The set of all integers x from a complete residue system (mod n) k such that (x, n )k = 1iscalled a k-reduced residue system (mod n). The number of elements of a k-reduced residue system (mod n) is denoted by ϕk(n).Clearly, ϕ1 ≡ ϕ.More generally, Cohen proves:

~~ϕk(n) = Jk(n)(4)~~

Formany other identities involving Jk, see [103], Chapter 4; and R. Sivaramakr- ishnan [421]. For new proofs of (3) and (4), see [79], [80]. P. Haukkanen [184] further generalized ϕk(n) and Jk(n) by introducing k the function ϕk,u(n) = number of u-tuples x1, x2,...,xu (mod n ) such that k (gcd(x1,...,xu), n )k = 1. One has ϕ1,u(n) = Ju(n); ϕk,1(n) = ϕk(n).Infact (see [184]) ku ϕk,u(n) = d µ(n/d), (5) d|n where µ is the classical Mobius¨ function. For a generalization of ϕk,u to the setting of Narkiewicz’s regular convolution, see [185]. Another generalization is the Jordan- Nagell totient of 2.3.3. As we have mentioned in section 3.1 (see relation (42’)), in 1999 T. Shonhiwa has considered the generalized totient

~~m( ) = { ≤ ,..., ≤ ,( ,..., , ) = }, Jk n card 1 a1 an n a1 ak m 1 which for m = n gives Jk(n).Then n k m( ) = µ( ) ( ) Jk n d 1 d|m d which for m = n yields relation (1).~~

~~2 Schemmel, Schemmel-Nagell, Lucas-totients~~

In 1869 V. Schemmel (see [103], p. 147) introduced the function Sk(n),asthe number of sets of k consecutive numbers each less than n, and relatively prime to n. It is known that k Sk(n) = n 1 − (6) p|n p

~~A generalization of Schemmel’s totient is the Schemmel-Nagell totient θs(M, N) of 2.3.3 from Chapter 2. For Schemmel’s totient, see also 3.1.4, 3.2.8, 3.3.3 of Chapter 3.~~

~~276 THE MANY FACETS OF EULER’S TOTIENT~~

In 1891 E. Lucas (see [103], p. 147) obtained a generalization of Schemmel’s totient. Let e1, e2,...,ek be any integers, and denote by ψk(n) the number of integers h, chosen from 0, 1, 2,...,n − 1such that h − e1, h − e2,...,h − ek are relatively prime to n. Then ψk is multiplicative, and λ ψk(n) = n 1 − (7) p|n p where λ is the number of distinct residues of e1,...,ek (mod p).Fork < n, e1 = 0, e2 =−1, ... , ek =−(k − 1) we reobtain Schemmel’s totient. Schemmel’s and Lucas’ totient functions have been further generalized by K. Nageswara Rao [309], in the setting of Cohen’s relatively k-primality (see (4)). Let (s)( ) s Sk n be the number of sets of k consecutive integers each less than n and which are s-prime to ns. Then ( ) k S s (n) = ns − ( ) k 1 s 8 p|n p (1) ≡ Remark that Sk Sk.Asimilar generalization to (7) holds true.

3 Ramanujan’s sum An important generalization of Euler’s totient is the Ramanujan sum C(n, r) deﬁned by C(n, r) = e(nx, r)(9) (x,r)=1,1≤x≤r where n ∈ Z, r ∈ N and e(a, b) = e2πia/b.Infact, Ramanujan’s sum is a common generalization of Euler’s totient ϕ and Mobius’¨ function µ,since

~~C(0, r) = ϕ(r), C(1, r) = µ(r)(10)~~

Ramanujan’s sum, with certain generalizations or analogues sums occured in 2.3.6 of Chapter 2, as well as in 3.1.5, 3.5.3, 3.6.1, 3.6.2 of Chapter 3. Many other generalizations of Ramanujan’s sum, due to E. Cohen, K. Nageswara Rao, M. Sugunamma, M. V. Subbarao - V. C. Harris, A. C. Vasu, L. Berardi - M. Cerasoli - F. Eugeni, etc. are included (with References) in [184]. R. Sivaramakr- ishnan, J. Hanumanthachari, or J. Hanumanthachari - V. V. Subrahmanya Sastri obtained Ramanujan sums for square reduced or k-free integers. V. Sita Ramaiah, P. J. McCarthy, P. Haukkanen have generalized Ramanujan’s sum in the setting of regular A-convolutions. We note that the unitary analogue of C(n, r) originates to E. Cohen. See also [69]. For unitary analogues of generalized Ramanujan sums, see also K. R. Johnson [213]. For Ramanujan sums on semigroups, see A. Grytczuk [159].

~~277 CHAPTER 3~~

Forgeneralizations of Ramanujan’s sum to matrices, see V. C. Nanda [312]. He studied also general arithmetic functions of integer matrices [313]. See also G. Bhowmik [43] and G. Bhowmik and O. Ramare[´ 44] on average orders of certain arithmetic functions of integer matrices (among others, the Euler totient of matrices). There is a vast literature on the Ramanujan expansions of arithmetic functions, i.e. representations in the form ar C(n, r).For a survey of results, see e.g. W. r Schwarz [405]. See also [159].

~~4 Klee’s totient~~

~~In 1948 V. L. Klee deﬁned the function Tk(n) as the number of integers 1 ≤ h ≤ n such that (h, n) is kth powerfree. Then n Tk(n) = µk(d) (11) d|n d~~

1/k where µk is Klee’s Mobius¨ function, given by µk(n) = µ(n ) if n is a kth power; 0, otherwise. See 2.3.5 of Chapter 2, where Suryanarayana’s generalization is included, too. It can be proved that (see e.g. [310], along with other identities) 1 T (n) = n 1 − (12) k pk pk |n

~~U. V. Satyanarayana and K. Pattabhiramasastry [390] proved that the average order of Tk(n) is n2 T (m) = + O(n n)() k ζ( ) log 13 m≤n 2 2k~~

A. C. Vasu [467] deﬁned an extension Tk,l of Klee’s totient as the number of ordered sets of l integers less than n which have with n the greatest common divisor ϕ kth power free. See also Subbarao-Harris’ k,q of 2.3.6 of Chapter 2. t s n A generalization Tk,s,t given by Tk,s,t = (µk(d)) ,with application in d|n d the evaluation of certain inﬁnite sums, can be found in B. C. Berndt [39]. For generalizations of Klee’s totient in the setting of A-convolutions, with many particular cases, see [184].

5 Nagell’s, Adler’s, Stevens’, Kesava Menon’s totients Nagell’s totient function θ(n, r) was introduced in 1923 by T. Nagell [307] (see also [81]) as the number of integers x (mod r) such that (x, r) = (n − x, r) = 1. In

~~278 THE MANY FACETS OF EULER’S TOTIENT fact, θ(n, r) can be considered as the number of solutions of the congruence~~

~~x + y ≡ n (mod r)(14) under the restrictions (x, r) = (y, r) = 1. E. Cohen [81] proved that µ(d) θ(n, r) = ϕ(r) ( ) ϕ( ) 15 d|r,(d,n)=1 d~~

θ(n, r) is multiplicative in both n and r, and another identity is: θ(n, r) = (−1)ω(d)ϕ(r/d); (16) d|r,(d,n)=1 see [421]. As a generalization of (14), let Ns(n, r) be the number of solutions of the congruence x1 + x2 + ···+xs ≡ n (mod r)(17) under the restriction (x, r) = 1, where x = (x1,...,xs). Then (see E. Cohen [82]), r s N (n, r) = J (g), with g = (n, r)(18) s g s In 1948 H. L. Alder [4] deﬁned a totient ϕ(n, r) as the number of ordered pairs x, y such that

~~x + y = n + r,(x, r) = (y, r) = 1 and 1 ≤ x ≤ r (n ≥ 0)~~

~~Clearly ϕ(0, r) = ϕ(r).Now, ε (p) ϕ(n, r) = r 1 − n ,(19) p|n p where 1, if p|n ε (p) = n 2, otherwise~~

~~Remark that ϕ(1, r) = S2(r), the Schemmel totient of order 2. Also, by (14),~~

~~ϕ(n, r) = θ(n, r) for all n, r (20)~~

Forageneralization of Nagell’s totient function and Ramanujan’s sum, see [177]. Forarecent generalization, with combinatorial applications, see [446]. An extension of Nagell’s (and Alder’s) totient has been introduced by H. Stevens [440]. Let F ={f1(x), ..., fk(x)} (k ≥ 1) be a set of polynomials with integral

~~279 CHAPTER 3~~

coefﬁcients, and let A be the set of all ordered k-tuples of integers a1,...,ak such that 0 ≤ a1,...,ak ≤ n. Let ϕF,k(n) be the number of elements in A such that r ( ( ), ..., ( ), ) = = a j gcd f1 a1 fk ak n 1. If n p j is the prime factorization of n, then j=1 r ... k N1 j Nkj ϕ , ( ) = − ( ) F k n n 1 k 21 j=1 p j where Nij is the number of incongruent solutions of fi (x) ≡ 0 (mod p j ). The Stevens totient is multiplicative, and contains, as special cases the Jordan totient Jk(n)(f1(x) = ··· = fk(x) = x); the Schemmel totient St (n) (k = 1, f1(x) = x(x + 1)...(x + t − 1)); the Nagell totient θ(n, r) = ϕF,1(r) (k = 1, f1(x) = x(n − x)). Let now ϕl,k(n) with 0 ≤ l < k be the numbers of k-tuples a1,...,ak with l ≤ ak ≤ n such that (al+1,...,ak, n) = 1. This is the Cashwell-Everett totient (see B. Rizzi [353]. By letting f1(x) = ···= fl (x) = nx,

~~fl+1(x) = ···= fk(x) = x, in (21), one gets l k p ϕ , (n) = n − ( ) l k 1 k 22 p|n p~~

Clearly, ϕ0,k(n) = Jk(n), Jordan’s totient. Forgeneralized Ramanujan sums connected with Stevens’ totient, see also L. Berardi and B. Rizzi [37]. For an asymptotic formula for ϕF,k, see L. Toth´ and J. Sandor´ [460]. In the case of a single polynomial f , with integral coefﬁcients, in 1967 P. Ke- sava Menon [222] considered the number of integers ϕ f (n) of x (mod n), such that ( f (x), n) = 1. Let D(s, d, n) ={s, s + d,...,s + (n − 1)d} be an arithmetic progression with (s, d) = 1, and let ϕ(s, d, n) denote the number of elements x ∈ D(s, d, n) such that (x, n) = 1. This function has been introduced by P. G. Garcia and S. Ligh [145]. This is a special case of ϕ f (n) for f (x) = s + (x − 1)d.L.Toth´ [457] obtained the asymptotic formula

~~3d2 ϕ(s, d, n) = x2 + O(x log x)(23) π 2 ( ) n≤x J d An asymptotic formula for ϕ(s, d, n) wasgiven by T. Maxsein [287]. d≤x~~

~~280 THE MANY FACETS OF EULER’S TOTIENT~~

Acombination of generalizations from [184] and Kesava Menon’s totient ϕ f is given in [458], where asymptotic results more general than (23) are also deduced. D. Goldsmith [149] considered a generalization of ϕ f (n) as follows. For each divisor d of n,letϕ f,d (n) be the number of 0 ≤ x < n such that ( f (x), n) = d. Clearly, ϕ f,1(n) = ϕ f (n).Let(n, r) = 1 and d|n, e|r. Then

~~ϕ f,ed(nr) = ϕ f,d (n)ϕ f,e(r)~~

~~For d = e = 1, this contains the multiplicative property of ϕ f .~~

6 Unitary, semi-unitary, bi-unitary totients The unitary analogue of ϕ(n) was introduced by E. Cohen [83] as follows. Let ∗ (a, b) denote the greatest divisor of a which is a unitary divisor of b (a divisor r of b b is called unitary, if r, = 1). If (a, b)∗ = 1, then a is said to be semi-prime to r b. Let ϕ∗(n) be the number of positive integers r ≤ n,semi-prime to n.Infact, ϕ∗(n) = dµ∗(n/d) = (pα − 1), (24) d|n pα n where µ∗(m) = (−1)ω(n) is the unitary Mobius¨ function. For unitary divisors see also 1.9 of Chapter 1, 2.2.2 of Chapter 2, and 3.6.1 of Chapter 3. For the corresponding notions of bi-unitary divisors and convolution, see 1.9 of Chapter 1, and 2.2.3 of Chapter 2. Adivisor d of n is called semi-unitary divisor (see J. Chidambaraswamy [73]) if d is semi-prime to n/d.For various sum evaluations on semi-unitary divisors, see [73]. The unitary analogue of Nagell’s totient has been introduced by J. Morgado [299]: θ ∗(n, r) is the number of integers x such that (x, r)∗ = (n − x, r)∗ = 1, 1 ≤ x ≤ r. θ ∗ is multiplicative in r,andis given by α α θ ∗( , ) = ( i − ) ( j − )() n r pi 1 p j 2 25 α α p i |n j i p j n

α α = 1 ... m where r p1 pm is the prime factorization ofr. m | θ ∗( , ) = ϕ∗( ) = − 1 If r n, then n r r r 1 αi . i=1 pi For identities involving θ ∗,see [300]. ∗( ) The unitary analogue of Schemmel’s totient is the number Sk n of k consecutive numbers x, x + 1,...,x + k − 1 with 1 ≤ x ≤ n and all semi-prime to n. This has been introduced by J. Morgado in [301].

~~281 CHAPTER 3~~

α α α α α α ∗ Let n = p 1 ...p m , and p = min{p 1 ,...,p m }.Ifk ≥ p , then S (n) = 0. 1 m 1α αm k ∗ ∗( i ) = i − The function Sk is multiplicative, and Sk pi pi k,so   1, if n = 1 ∗( ) = , > ≥ α ( ) Sk n 0 if n 1 and k p 26  α α α ( 1 − )...( m − ), > ≤ < p1 k pm k if n 1 and 1 k p

~~For the unitary analogues of certain remarkable identities (e.g. of Brauer- Rademacher’s, Landau’s, or Anderson-Apostol’s), see P. J. McCarthy [72].~~

~~7 Alladi’s totient~~

The k-ary divisors have been considered e.g. in 2.3.5 of Chapter 2 and in relation with (11). Another notion of higher order divisors is due to K. Alladi [5]. A divisor d of n is called of ﬁrst order,innotation d|1n. Let (a, b)1 denote the largest divisor of adividing b. When (a, b)1 = 1, we say that a is prime to b order 1. A divisor d of n n is said to be of second order,denote d|2n,if , d = 1. Let (a, b)2 denote the d 1 largest divisor c of a satisfying c|2b.If(a, b)2 = 1wesay a is prime to b order 2. A n divisor d of n is a divisor of third order, denoted d|3n,if , d = 1. Let (a, b)3 be d 2 the largest divisor c of a that satisﬁes c|3b.If(a, b)3 = 1, we say a is prime to b order n 3. By generalizing, we say that d|r n if , d = 1 and (a, b)r = max{c|1a, c|r b}. d r−1 If (a, b)r = 1, then a is prime to b order r. Let ϕr (n, x) = number of 1 ≤ a ≤ x such that (a, n)r = 1beAlladi’s totient, ϕ ( ) = ϕ ( , ) ϕ ( , ) = ≤ ≤ and put r n r n n .Similarly, let r n x number of 1 a x with ( , ) = ϕ ( ) = ϕ ( , ) ( ) > Fr−1 n a r 1. Let r n r n n .Let fr x be the least integer x, when r Fr Fr−1 is odd, and ≥ x,when r is even, where (Fr ) is the Fibonacci sequence given Fr α α = = = + ( ≥ ) = 1 ... m by F0 0, F1 1, Fn Fn−1 Fn−2 n 2 . Let n p1 pm be the prime factorization of n. Then m 1 ϕr (n) = n 1 − (27) fr (αi ) i=1 pi

~~An interesting consequence is that~~

~~ϕ1(n) ≤ ϕ3(n) ≤ ϕ5(n) ≤ ···≤ϕ6(n) ≤ ϕ4(n) ≤ ϕ2(n)(28)~~

~~282 THE MANY FACETS OF EULER’S TOTIENT~~

k k n Let σ , (n) = d , σ (n) = .Then r k r,k d d|r n d|r n n k+1 k+1/2 σr,k(m) = cr,kn + O(n ) m=1 (29) n σ ( ) = k+1 + ( k+1/2) r,k n cr,kn O n m=1 where cr,k and cr,k are certain constants depending only on r and k.

8 Legendre’s totient Afamous generalization of Euler’s totient is the Legendre totient function ϕ(x, n) = number of positive integers ≤ x, which are relatively prime to n, introduced by Legendre in 1794 (see [103], p. 115). Clearly, ϕ(n, n) = n.Legendre proved essentially that x ϕ(x, n) = µ(d) (30) d|n d Relation (30) immediately implies ϕ(n) ϕ(x, n) = x + O(d(n)), (31) n where d(n) is the number of divisors of n. This gives rise to the function xϕ(n) (x, n) = ϕ(x, n) − . n In 1946 P. Erdos¨ [120] proved that |(x, n)| > c · 2ω(n)/2/(log ω(n))1/2,(32) while D. H. Lehmer [249] showed that |(x, n)| < 2ω(n)−1,(33) where ω(n) denotes the number of distinct prime factors of n.Itcan be conjectured that (see [120]) |(x, n)|=O(2ω(n)), if ω(n) →∞ (34) Arelated result is the following (see [251]): ω(n) µ(d) ≤ ( ) ω( )/ 35 d|n,a≤d≤b [ n 2]

~~283 CHAPTER 3~~

In 1974 D. Suryanarayana [246] obtained the following two inequalities: 1 ϕ(n) 1 (x, n) + {x}− ≤ (d(n) − ψ(n)/n); x ≥ 1, n ≥ 2,(36) 2 n 2 ψ(n) d(n) ϕ(n) d(n) ψ(n) − ≤ (x, n) +{x} ≤ − + 1, for x ≥ 1, n ≥ 2,(37) n 2 n 2 n such that ([x], n) = 1. Here ψ(n) is the Dedekind arithmetical function (see [103], p. 123) given by 1 ψ(n) = n 1 + (p prime). p|n p For Dedekind’s function, see e.g. E. Cohen [84], D. Suryanarayana [449], J. Sandor´ [369] and J. Sandor´ and R. Sivaramakrishnan [383]. A. Sivaramasarma [425] stated that if x ≥ 1, n ≥ 2, and m = (n, [x]), then ϕ(n) 1 1 d(n) d(m) md(m)ψ(n) (x, n) +{x} − ≤ + − ,(38) n 2 m 2 2 nψ(m) where {x}=x − [x]isthe fractional part of x. The unitary analogue of Legendre’s totient has been introduced by Cohen in [83] as the number ϕ∗(x, n) of positive integers a ≤ x such that (a, n)∗ = 1. In particular, ϕ∗(n, n) = ϕ∗(n).Amore general function, considered by J. Chidambaraswamy [73] is ϕ∗( , ) = r ( ≥ ) r x n a r 0 a≤x,(a,n)∗=1 He proved that

xr+1 ϕ∗(n) ϕ∗(x, n) = · + O(xr d∗(n)), (39) r r + 1 n for any r ≥ 0, x ≥ 1, n ≥ 1; where d∗(n) denotes the number of unitary divisors of n. P. Haukkanen [186] obtained the unitary analogue of (38): ϕ∗( ) ( ) ( ) ( )ψ∗( ) ∗ n 1 1 d n d m md m n (x, n) +{x} − ≤ + − (40) n 2 m 2 2 nψ∗(m) 1 m = ( x , n)∗ ψ∗(n) = σ ∗(n) = n + where [ ] , 1 a is the unitary analogue pa n p of Dedekind’s function (in fact, the sum of unitary divisors of n), and ∗(x, n) = ϕ∗(x, n) − xϕ∗(n)/n.

~~284 THE MANY FACETS OF EULER’S TOTIENT~~

In fact, Haukkanen works in the context of A-convolutions, and (38), (40) are particular cases. Let S ⊂ N be a non-void set of positive integers. The Legendre totient of u arguments associated to S was introduced by E. Cohen [191] as ϕS(x1,...,xu; n) = number of u-tuples a1,...,au with a j (mod x j )(j = 1, u) such that ((a1,...,au), n) ∈ S. When S ={1},andu = 1, this reduces to the classical Legendre function. Then x1 xu ϕS(x1,...,xu; n) = µS(d) ... ,(41) d|n d d where µS is the Mobius¨ function of the set S (see 2.2.4 of Chapter 2, where the more general µS,A is introduced, too). For certain arithmetical products, with asymptotic expansions, which involve also ϕS,seeJ. Sandor´ and L. Toth´ [386]. In [184] this function is generalized to Ak-convolutions, so (41) becomes ϕ ( ,..., ; ) = µ ( ) / k ... / k ( ) S,A,k x1 xu n S,Ak d [x1 d ] [xu d ] 42 d∈Ak (n) Foranapplication of Legendre’s totient function ϕ(x, n) in the study of primes in a real quadratic ﬁeld, see A. Granville, R. A. Mollin and H. C. Williams [157], where it is proved e.g that if x ≥ 4.7 · 1015, then 1 π(x) ≤ ϕ(x, n), 3 where n is the product of all primes ≤ 14 log x, and π(x) is the number of primes ≤ x.

9 Euler totients of meet semilattices and finite fields Relation (35) of 3.6 gives a generalization of Euler’s totient to a finite meet semilattice, and a function f : P → C.In[191] P. Haukkanen and J. Wang considered set-valued maps f : P → A, where P is a meet-semilattice, S ⊂ P is a fixed set, and A is a collection of finite subsets of S. The mapping f is called a regular S-representation of P if f (x ∧ y) = f (x) ∩ f (y) for all x, y ∈ P. The Euler totient ϕ f (x) of P with respect to a regular S-representation f is defined as the number of elements z ∈ f (x) such that z ∈ f (y) for y < x (we assume that P is a meet-semilattice and that every principal order ideal of P is finite). Then ϕ f (x) = µ(y, x)| f (y)| (43) y≤x

~~285 CHAPTER 3~~

Examples. 1) When P is N, ordered by divisibility, and f (n) = Un, the set of all nth roots of unity, then with U = Un, f will be a regular U-representation of N. n≥1 Then ϕ f (n) = ϕ(n), the classical Euler totient. ( ) = = ϕ ≡ ϕ = 2) Let fk n Unk , U Un.Then fk k Cohen’s totient function. n≥1 ( ) = ( )k ϕ = = 3) For gk n Un one obtains gk Jk Jordan’s totient function. Let Fq denote a finite field of q elements, and let V be a vector space over Fq . Let A be the set of all finite-dimensional subspaces of V . Let fq : P → A be a regular V -representation of P, i.e. let

~~fq (x ∧ y) = fq (x) ∩ fq (y), ∀ x, y ∈ P (44)~~

~~ϕ ( ) ( ) The q-Euler function fq x is the number of 1-dimensional subspaces of fq x that are not contained in fq (y) for y < x. Then~~

~~( ) qdim fq y − 1 ϕ (x) = µ(y, x) (45) fq − y≤x q 1~~

~~E.g., for example 1) above, when fq (n) = Fq (Un),wegettheq-analogue of the classical Euler totient:~~

~~qd − 1 ϕ (n) = ϕ (n) = µ(n/d) (46) q fq − d|n q 1~~

~~The q-analogue of Jordan’s totient Jk (see example 3) above) will be~~

~~qdk − 1 ϕ , (n) = ϕ (n) = µ(n/d) (47) J q gq,k − d|n q 1~~

Let f be a regular representation of P, and α : P → G, where (G, +) is an Abelian group. The Ramanujan sum of P with respect to f and α is deﬁned by C f,α(x) = α(z), (48) z∈S f (x) where S f (x) ={z ∈ f (x) : z ∈ f (y) for y < x}. One has C f,α(x) = µ(y, x) α(z), (49) y≤x z∈ f (y) and the q-analogue of the classical Ramanujan sum can be introduced (see [191]).

~~286 THE MANY FACETS OF EULER’S TOTIENT~~

10 Nonunitary, infinitary, exponential-totients We now consider totient functions suggested by certain notions of Chapters 1, 2. In 1.9 of Chaper 1, as well as 2.2.6, 2.2.10 of Chapter 2 there are considered the nonunitary, infinitary, exponential divisors, and convolution products. See also 1.13 and 1.14. The nonunitary totient function ϕ#(n) is defined as the number of positive integers ≤ n which are not divisible by any of the nonunitary divisors of n.Ifn = n·n#, where n is the squarefree, while n# the powerful part, of n, then

~~ϕ#(n) = n · ϕ(n#)(50)~~

One has also ϕ#(d) = σ ∗(n), (51) d|n where σ ∗(n) is the sum of unitary divisors of n (see [257]). Similarly,     ϕ#( ) = σ( ) σ ∗( #) − ( e − e−1 + ) ( ) d n  n p p 1  52 d|#n pe n# It is conjectured that if ϕ#(d) = n, then n# is even. d|n In part 2.2.10 of Chapter 2 there are introduced the infinitary divisors and Mobius’¨ infinitary function. If n = p1 p2 ...pt , where p1 < p2 < ··· < pt are the I - components of n, then one defines the infinitary totient function by t 1 ϕ∞(1) = 1,ϕ∞(n) = n 1 − (n > 1)(53) j=1 p j

~~The function ϕ∞ is I -multiplicative, i.e. (m, n)∞ = 1(i.e. greatest common in- ﬁnitary divisor is =1), then ϕ∞(mn) = ϕ∞(m)ϕ∞(n). Similarly, ϕ∞(d) = n (54) d|∞n~~

For these results and what follow, see [87]. Let also ϕ∞(x, n) be the inﬁnitary Legendre totient deﬁned by ϕ∞(x, n) = number of a ≤ x such that (a, n)∞ = 1. One has the following asymptotic results:

~~ε ε ϕ∞(x, n) = k∞(n)x + O(n x )(55)~~

287 CHAPTER 3 for any x > 0, n ≥ 1, (ε > 0),where the multiplicative constant implied by the big of notation depends only on ε, and k∞ is an arithmetic function deﬁned by k∞(1) = 1, t p j k∞(n) = + . Similarly, j=1 p j 1

~~2 x 1+ε ϕ∞(n) = · A + O(x log x)(56) n≤x 2~~

∞ 2 µ∞(n)k∞(n) for any x > 1, and any ε>0. Here A = .For the function k∞(n) 2 n=1 n one has: ε k∞(n) = Cx + O(x log x)(57) n≤x 1 for all x > 1, where C = 1 − . ( + )2 P∈I P 1 For the exponential divisors and convolutions, see 1.10 of Chapter 1, and 2.2.6 of Chapter 2. The exponential totient function is introduced in [270] of Chapter 2 (classical case), and in [486] of Chapter 2 (regular convolutions). Let ϕe(n) denote the number of all a ≤ n with (a, n)e = 1, and put ϕe(1) = 1. If = a1 ... ar > n p1 pr is the prime factorization of n 1, then

~~ϕe(n) = ϕ(a1)...ϕ(ar )(58)~~

~~ϕe is a multiplicative function, and (see [380]) ϕe(d) = a1 ...ar (59) d|en~~

Other properties of ϕe are: If n|em, then ϕe(n)|ϕe(m); ϕe(n)de(n) ≥ a1 ...ar ;if all ai (i = 1, r) are odd, then ϕe(n)de(n) ≥ σ(a1)...σ(ar ), where de(n) is the sum of exponential divisors of n; log log n log 4 lim sup(log ϕe(n)) = (60) n→∞ log n 5

Finally, note that the Golomb-Guerin totient function ϕ∇ is introduced in 2.2.8 of Chapter 2 (see [165]). For generalized Euler functions involving rational arithmetic functions, and the extensions of the Fleck-Popoviciu-Buschman-Hsu-Wang Mobius¨ function, see 2.2.1 and 2.2.4 of Chapter 2. For a Lucas-sequence based generalization of Euler’s totient, see 3.2.2 of Chapter 3.

~~288 THE MANY FACETS OF EULER’S TOTIENT~~

11 Thacker’s, Leudesdorf’s, Lehmer’s, Golubev’s totients. Square totient, core-reduced totient, M-void totient, additive totient In 3.4.1 we have introduced Thacker’s totient function, as well as the generalized totient due to Leudesdorf. Lehmer’s totient function is considered in 3.4.2.In 3.2.5 there is a class F1 of generalized totient functions due to Siva Rama Prasad and Fonseca. Other functions are the Golubev totient functions. In 1953 V. A. Golubev [151], by studying twin primes and other special types of primes, deﬁned ϕ2(n) = number of pairs (a1, a2) of positive integer such that a2 − a1 = 2, (a1, n) = 1, a1 ≤ n. Then ϕ2 is multiplicative, and   2  n 1 − , if n is odd   p|n p ϕ ( ) = ( ) 2 n 61   n 2  1 − , if n is even 2 p|n p

~~If ϕ2k(n) = number of pairs (a1, a2) with a2 − a1 = 2k, (a1, n) = 1, a1 ≤ n, then ϕ2k is multiplicative, and 1 2 ϕ (n) = n 1 − 1 − (62) 2k q q q|n,q|2k q|n,q2k~~

(m) More generally, let ϕ (n; r1,...,rm−1) = number of sets of positive integers a1,...,am each relatively prime to n, with given differences ak − ak−1 = rk−1 (1 ≤ k ≤ m − 1), all of the r’s being even, and a1 ≤ n. Then (m) 1 1 ϕ (n; r1,...,rm−1) = n 1 − ... 1 − (63) p1 pm p1,...,pm in which pk denotes the prime divisors of n such that the set {0, r1, r1 + r2,...,r1 + r2 +···+rm−1} contains at most j(pk; r1,...rm−1) incongruent elements (mod pk). (64) See also R. G. Buschman [58]. The square-totient of R. Sivaramakrishnan [422] is introduced as follows. Let n ≥ 1, and consider the set of integers a (mod n) such that (a, n) = a square ≥ 1, and contained in a complete residue system (mod n) is called a ”square-reduced residue system (mod n)”. The number of elements in a square-reduced residue system (mod n) is denoted by b(n).

~~289 CHAPTER 3~~

Let 1, if n is a perfect square ε(n) = 0, otherwise Then n b(n) = ε(d)ϕ (65) d|n d The function b is multiplicative, and satisﬁes certain identities, see e.g. [421]. Let γ(n) denote the product of distinct prime divisors of n > 1 (i.e. ”core of n”), and put γ(1) = 1. Let S denote a complete residue system (mod n), and let M be the set of integers c (mod n) such that (c, n) = γ(n),and contained in S. This subset M will be called as the ”core-reduced residue system (mod n)”. The core-reduced totient of J. Sandor´ and R. Sivaramakrishnan [383] is the cardinality m(n) of M. The function m(n) is multiplicative, and n m(n) = ϕ(s), where s = (66) γ(n)

Let c ≡ 0 (mod n).Ifc (mod n) is such that (c, n) = γ(n), then ck ≡ 0 (mod n) for some integer k ≥ 1. Therefore, the elements of a core-reduced residue system (mod n) are nonzero nilpotent elements in the ring Zr .Infact,m(n) does not count all the nilpotent elements in Zr ,but the number of nilpotents elements in n Z is s − 1, where s = (see [383]). n γ(n) Let M be a set of positive integers with min M = s ≥ 2. A positive integer r = ai ∈ = , n pi is said to be M-void,ifai M, i 1 r. Let Q M denote the set of all i=1 M-void integers, and denote by qM its characteristic function. Let λM be deﬁned by qM (n) = λM (d).Itisclear that qM and λM are multiplicative functions, and for d|n all prime powers pa,   −1, if a ∈ Ma− 1 ∈ M λ ( a) = , ∈ − ∈ ( ) M p  1 if a Ma 1 M 67 0, otherwise

M. V. Subbarao and R. Sitaramachandrarao [444] deﬁne the M-void analogue of the Euler totient function by the number ϕM (n) of integers x ≤ n such that (x, n) ∈ Q M . (This is a particular case of the Cohen totient ϕS with S = Q M ). We have n ϕM (n) = λM (d) (68) d|n d

~~290 THE MANY FACETS OF EULER’S TOTIENT and a ≤ ≤ − ϕ ( a) = p if 1 a s 1 M p a s a−s a p + λM (p )p + ···+λM (p ) if a ≥ s~~

~~Let A(x,ϕM ) be the number of positive integers n such that 1 ≤ ϕM (n) ≤ x. Then, for each c > 0, as x →∞,~~

c A(x,ϕM ) = A(ϕM )x + O(x(δ(x)) )(69) where ∞ −s −1 a −1 A(ϕM ) = 1 − p + (1 − p ) (ϕM (p )) pprime a=s and 3 −ε δ(x) = exp{−(log x) 8 }. For particular sets M one can reobtain the sets of r-free integers, (k, r)-integers, unitary r-free integers, semi-r-free integers, etc. By counting rational points on a projective space or on quadratic twists of an elliptic curve, A. Sato [389] introduced the following generalizations of Euler’s totient. Let q > 0beagiven integer, and put ϕq (x) = number of 0 < y ≤ x such that (x, qy) = 1; ψq (y) = number of 0 < x ≤ y such that (x, qy) = 1. Clearly, ϕ1(x) = ϕ(x), ϕ1(y) = ϕ(y). The following asymptotic formulae are due to Shigeki Akiyama (see [389]): 2 ϕq (x) = Cq B + O(B log B) x≤B (70) 2 ψq (y) = Cq B + O(B log B), as B →∞ y≤B

r 3 pi where Cq = , with (pi ) the distinct prime divisors of q. π 2 + i=1 pi 1 Finally, we note that the additive analogue of Euler’s totient has been deﬁned by r = ai > K. Atanassov [18] as follows: For n pi (prime factorization) and n 1, let i=1

r − ρ( ) = ai 1( − )() n pi pi 1 71 i=1 Put ρ(1) = 0. Then clearly ρ is an additive function, i.e. ρ(nm) = ρ(n) + ρ(m) for all (m, n) = 1. The function ρ is connected also to other functions, e.g. to the

~~291 CHAPTER 3 additive analogue of the σ (sum-of-divisors) function, deﬁned by~~

~~+ r pai 1 − 1 θ(n) = i ,θ() = .() − 1 0 72 i=1 pi 1~~

~~One has ρ(n) ≤ ϕ(n); θ(n) ≤ n − 1forall n ≥ 1 (73) and ρ(n) ≥ r(ϕ(n))1/r ; θ(n) ≥ r(σ (n))1/r for n > 1.(74)~~

12 Euler totients of arithmetical semigroups, finite groups, algebraic number fields, semigroups, finite commutative rings, finite Dedekind domains Generalized integers and arithmetical semigroups have been considered in 2.2.11 and 2.5.1 of Chapter 2. If G is an arithmetical semigroup and H an arithmetical subsemigroup of G,forxi ∈ G (1 ≤ i ≤ k) one can define the H-greatest common divisor of the xi ’s by (x1,...,xk)H = h if h is an element from H with the largest possible norm that divides each xi (1 ≤ i ≤ k). Let ϕk(H, x) denote the number of k-tuples x1,...,xk of elements of G such that |xi |≤x (i = 1, k) and (x1,...,xk)H = 1. H. Wegman [479] called an arithmetical semigroup δ-regular, if its counting function G(x) satisfies G(x) = xδ L(x), where L(x) is defined for all x > 0, and L(ax) → 1asx →∞for every a > 0. If G is δ-regular, then Wegman proved L(x) −s that its zeta functions ζG (s) = |n| converges for s >δand diverges for s <δ. n∈G tδ G(x/t) Let r(x, t) = − 1, for x, t > 0. The following result is due to S. Porubsky´ G(x) [339]. Let G be δ-regular such that r(x, t) is uniformly bounded for all x, t > 0. −kδ Let H be an arithmetical subsemigroup such that the series |h| = ζH (kδ) h∈H converges. Then 1 k ϕk(H, x) = + o(1) (G(x)) (75) ζH (kδ) If G satisfies the axiom: there exist positive constants A and δ and a constant η with ≤ η<δsuch that

~~G(x) = Axδ + O(xη) as x →∞ (76)~~

292 THE MANY FACETS OF EULER’S TOTIENT then for k ≥ 2, 1 ( (k−1)δ+η) > , = η> ϕ ( , ) = k kδ + O x if k 2 or k 2 and 0 ( ) k H x A x ( δ ) = ,η= 77 ζH (kδ) O x log x if k 2 0 In 2.5.3 there were introduced Mobius¨ functions and Euler functions of finite groups, soluble groups, algebraic number fields, or algebraic function fields (in fact, having introduced a Mobius¨ function, the Euler functions follow at once). We note here that L. Weisner [477] computed the Mobius¨ and Euler functions of a finite p- group as follows. Recall that the Frattini subgroup of a finite group G (denoted by Frat G)isthe intersection of all maximal, proper subgroups of G.IfG is a p-group, then Frat G = [G, G] · G p,bythe Burnside basis theorem. Now, Weisner proved that the Mobius¨ function of a finite p-group G is given by (−1)d pd(d−1)/2, where pd = [G : H]ifFrat G ≤ H, µ(H) = (78) 0, if Frat G H Now set pr =|G| and ps =|G : Frat G|. The Euler function of G is then given by s−1 (r−s)n n i ϕn(G) = p (p − p )(79) i=0 Forarecent account√ on the Euler functions of finite groups, see K. S. Brown [55]. Let K = Q( D) be an arbitrary (real or complex) quadratic field; let a denote an integral ideal in the ring R of algebraic integers in K (a = (0)) with norm N(a) (i.e. the number of residue classes of R modulo a). Let ϕ(a) be the number of those residue classes which are prime to a (see e.g. E. Hecke [196]). Then it is well known that ϕ(a) = N(a) (1 − N(p)−1) (p a prime ideal), and p|a 2 ϕ(a) = kx + R0(x), (80) N(a)≤x ϕ(a) = kx + R (x), ( ) ( ) 2 1 81 N(a)≤x N a 1 = αρ x + λ + E (x), ( ) ϕ( ) log 0 82 N(a)≤x a N(a) = αρx + ρζ ( ) x + E (x)() ϕ( ) k 0 log 1 83 N(a)≤x a where ρ is the residue of the Dedekind zeta-function ζk(s) at s = 1; k = −1 ρ(2ζK (2)) ; α = ζK (2)ζK (3)/ζK (6); and λ is another constant. By generalizing

293 CHAPTER 3 results obtained in the classical case by Pillai and Chowla, W. G. Nowak [324] has obtained the following estimates: k 2 2 R0(n) = x + O(x δ(x)), (80 ) n≤x 2 R1(x) = kx + O(xδ(x)), (81 ) n≤x where δ(x) = exp{−c(log x)3/5(log log x)−1/5} and c > 0isaconstant depending on K . Similarly, by extending results on E0 and E1 from the classical case, Nowak has proved: 1 E0(n) = αρ − ζK (0)ρ log x + O(1)(82 ) n≤x 2 13/14 E1(n) = ωx + O(x )(83 ) n≤x where ω is a constant. We note that

4/3 1/3 −2/3 1/3 R0(x) = O(x ), R1(x) = O(x ), E0(x) = O(x ), E1(x) = O(x )(84) butinthe classical case the results are stronger (see [291], Chapter 1). For the Euler functions of semigroups, ﬁnite commutative rings, and ﬁnite Dedekind domains, see 3.2.2 of this Chapter.

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~~323 CHAPTER 3~~

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~~324 THE MANY FACETS OF EULER’S TOTIENT~~

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~~325 CHAPTER 3~~

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~~326 THE MANY FACETS OF EULER’S TOTIENT~~

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~~327 Chapter 4~~

~~SPECIAL ARITHMETIC FUNCTIONS CONNECTED WITH THE DIVISORS, OR WITH THE DIGITS OF A NUMBER~~

~~4.1 Introduction~~

There are many particular arithmetic functions, numbers or sequences connected with some important notions or results which appear in Number theory, and in fact all Mathematics. Some of them are non standard functions (such as d,σ,ϕ,µ,ω,,p, P etc.), and were studied in [260] or in former chapters of this book. The aim of this chapter is the study of some other functions which at one part are not so well-known, and are scattered in various fields of study, or at another part have not been previously included. The former one includes arithmetic functions connected to the prime factorization of a number, or the consecutive divisors (prime or not). The functions related to the digits of a number written e.g. in a decimal (or binary) scale constitute also an important field of study, with many applications. The divisors and the digits of a number have a strong connection, it is sufficient to only mention the congruence property n ≡ s(n)(mod 9), where s(n) denotes the sum- of-digits of n in decimal representation of n.

~~329 CHAPTER 4~~

~~4.2 Special arithmetic functions connected with the divisors of a number~~

~~1 Maximum and minimum exponents~~

~~= a1 ... ar > Let n p1 pr 1bethe prime factorization of n, and put~~

H(n) = max{a1,...,ar }, h(n) = min{a1,...,ar } (1) and H(1) = h(1) = 1. These functions (called as the maximum and minimum exponents in factoring) occured also in Chapter III, when extending the Euler divisibility theorem. In 1969 I. Niven [265] proved that H(n) = c0x + RH (x)(2) n≤x ∞ −1 where c0 = 1 + (1 − ζ(k) ), and RH (x) = o(x) and that k=2 h(n) = x + Rh(x)(3) n≤x √ √ where Rh(x) = c x + o( x), with c = ζ(3/2)/ζ(3) (where ζ is the Riemann zeta function) conjectured previously by P. Erdos¨ (see [265]). We note here that the generating functions of max{n1,...,nk} and min{n1,...,nk} (where a1,...,ak are arbitrarily nonnegative integers) are deduced by L. Carlitz [52]. For example ∞ x x ...x { ,..., } n1 ... nk = 1 2 k ( ) min n1 nk x1 xk 4 (1 − x1)...(1 − xk)(1 − x1 ...xk) n1,...,nk =0 D. Suryanarayana and R. Sitaramachandra Rao [322] improved Niven’s results to

1/2 3/5 −1/5 RH (x) = O(x exp(−c(log x) (log log x) )) (5) and √ √ √ √ ( ) = + 3 + 4 + 5 + ( 1/6)() Rh x c x c1 x c2 x c3 x O x 6 A(x) where c > 0andc1, c2, c3 are constants. H. Cao [49] rediscovered (5), and a weaker form of (6). In another paper [50] he improved the exponent 3/5 to 3/5−ε.In[163] T. Gu and H. Cao assert without proof that

~~1/6 1/6 4/7 −3/7 Rh(x) = A(x) + C4x + O(x exp(−D(log x) (log log x) )(7)~~

330 SPECIAL ARITHMETIC FUNCTIONS but, according to A. Ivic´ [191] this was formerly established. These results can be further improved by using the strongest known form for the asymptotic formula on the number of squarefull integers not exceeding x. By applying the ideas used by Niven, S.ˇ Porubsky´ [278] has extended (2) and a weaker form of (3), to arithmetical semigroups (see e.g. Section 2.5 of Chapter II). Forexample, if G is an arithmetical semigroup with δ-regular G(x) (where G(x) = 1), then n∈G,|n|≤x 1 lim hG (m) = 1 (8) x→∞ ( ) G x |m|≤x,m∈G

( ) = { ,..., } = a1 ... ar ∈ where hG m min a1 ar , when m p1 pr G is the unique factorization of m into a product of generators. Let f : N → N and put M f ={n = 1, 2, ··· : f (n)|n}, i.e. the set of positive integers n such that f (n) divides n. The asymptotic density of the set M f is well known for some special functions, including d(n), ω(n), (n).In1994 A. Schinzel and T. Salˇ at´ [301] have proved that

d(Mh) = 1,(9) and ∞ α− α+ α− α 1 1 p ordp 1 − 1 1 p ordp − 1 d(M ) = + − H ζ( ) ζ(α + ) α+1 − ζ(α) α − 2 α=2 1 p|α p 1 p|α p 1 (10) where ord pα denotes the exponent of the prime p in the canonical representation of A(x) α. Here d(A) = lim denotes the asymptotic density of the set A. x→∞ x They proved also that the sequences (xn) and (yn),given by H(n) h(n) x = , y = (n = 2, 3,...) (11) n log n n log n 1 are dense in the interval 0, . log 2 In 1951 H. Fast [121] deﬁned the concept of statistical convergence.Ase- quence of real numbers (an)n≥1 is said to converge statistically to a ∈ R,denoted by limstat an = a, provided that for each ε>0wehave

~~d(Aε) = 0,(12) where Aε ={n : |an − a|≥ε}.~~

~~331 CHAPTER 4~~

There exists today a vast literature on statistical convergence of sequences, see e.g. J. Fridy [132], J. Fridy and C. Orhan [134], J. Connor [57], J. A. Fridy, M. I. Miller and C. Orhan [133]. Schinzel and Salˇ at´ [301] prove that H(n) h(n) limstat = limstat = 0 (13) log n log n It is interesting to note that there is a strong connection between the concepts of normal order of an arithmetic function and statistical convergence. Let f, F : N → R be two arithmetic functions. Then F is a normal order of f if and only if f (n) limstat = 1 (14) F(n)

Indeed, suppose F is a normal order of f .If0<ε<1, then there is a set Bε ⊂ N such that d(Bε) = 1, F(n)>0 for n ∈ Bε and (1 − ε)F(n)< f (n)<(1 + ε)F(n). From this we obtain f (n) − 1 <ε, n ∈ Bε. F(n) f (n) Therefore the inequality − 1 ≥ ε holds at most for all n ∈ N \ Bε = F(n) Aε,where d(Aε) = 0. Thus (14) holds. Conversely, if (14) holds, then it follows immediately that (1 − ε)Fn < f (n)<(1 + ε)F(n) for almost all n ∈ N (see e.g. G. H. Hardy and E. M. Wright [177]). Clearly, the constant function F(n) = 1 (n ∈ N) is a normal order of h.(15) Schinzel and Salˇ at´ [301] prove that H cannot have any non-decreasing normal order, i.e. if F is any non-decreasing function on N, then F os not a normal order of H.(16)

~~2 The product of exponents Now, similarly to (1) deﬁne~~

β(n) = a1a2 ...ar ,β(1) = 1 (17) In 1947 D. G. Kendall and R. A. Rankin [213] showed that 1 ζ(2)ζ(3) lim β(n) = = 1.943 ... (18) x→∞ ζ( ) x n≤x 6 In 1973 J. Knopfmacher [223] proved that for all ε>0,

~~1 (1+ε) log log n β(n)~~

~~332 SPECIAL ARITHMETIC FUNCTIONS~~

~~In fact, the Dirichlet series of β(n) is~~

~~∞ β(n) ζ(s)ζ(2s)ζ(3s) = (20) s ζ( ) n=1 n 6s ∞ gk(n) n Let dk = , where gk(n) = µ . n=1 n t|n,β(t)=k t Then E. Kratzel¨ [236] proved that 1 = (dk + o(1))h (21) x~~

~~m + β(m) ≤ n for every m < n (23)~~

~~In fact, the density of integers satisfying (23) is positive. More generally, let αi be the density of integers n for which~~

~~max{m + β(m) : m < n}=n + i (24) Then αi exists for every i, and αi = 1. i≥0 Denote by Si the set of integers m for which the number of solutions of equation~~

~~n + β(n) = m (25)~~

~~333 CHAPTER 4~~

is equal to i. Erdos¨ conjectures that the set Si has a density βi ≥ 0 and βi = 1. Here probably β0 > 0, but βi > 0isnotsure always. i≥0 The analogues questions for n +d(n), n +ϕ(n), n +σ(n),etc. are open problems. We note that G. E. Hardy and M. V. Subbarao [176] have introduced the so-called highly powerful numbers n as numbers satisfying β(n)>β(m) for all 1 ≤ m < n.

3 Arithmetic functions connected with the prime power factors We now consider other arithmetical functions related to the prime power factors of n. Let p be a prime. Let ap(n) = ord pn be the arithmetical function deﬁned by

ap(n) ap(1) = 0, for n > 1, p n,(26) ( ) ( )+ i.e. pap n |n but pap n 1 n. It is immediate that ap is completely additive, i.e. ap(nm) = ap(n) + ap(m) for all n, m ≥ 1. On the other hand, since log n a (n) ≤ (27) p log p it easily follows that the series a (n) p s > s ≤ .() s is convergent for 1 and divergent for 1 28 n≥1 n

Since ap is completely additive, one has n k ap(k) = ap(n!) = [n/p ], k=1 n≤bn log n where b = (Legendre’s theorem, see e.g. [177]); it follows easily that n log p a (1) + a (2) + ···+a (n) 1 lim p p p = (29) n→∞ n p − 1 ={ ( ) = } ( ) Let Tk n : ap n k andTk xdenote the number of elements of Tk which x   x  pk  are ≤ x. Since Tk(x) = −  ,from simple estimates we get pk p

~~p − 1 d(T ) = (30) k pk + 1~~

~~334 SPECIAL ARITHMETIC FUNCTIONS~~

where d is the asymptotic density. Related to the asymptotic density of the set Map (deﬁned in section 1 (see relation (9), (10)) one has the following result due to T. Salˇ at´ [296]: ( ) = ( − ) 1 + ( − ) 1 ,() d Map p 1 p 1 31 k+1 k−sk +1 (k,p)=1 kp (k,p)>1 kp where psk k.Asacorollary (see [296]) one gets

lim d(Ma ) = 0 (32) p→∞ p a (n) The sequence log p p is dense in (0,1); (33) log n n≥2 for a proof of O. Strauch, see [296]. For the statistical limit one can write: a (n) limstat log p p = 0 (34) log n r = ai If n pi is the prime factorization of n, deﬁne i=1 j−1 1 α γ ( ) = i , ≤ ≤ ( ) j n log pi 2 j r 35 log p j i=1

Put P(n) = max γ j (n). Then P. Erdos¨ [97] proved that 2≤ j≤r ( ) = ( + ( ))( )/( )() P n 1 o 1 log3 n log4 n 36 γ where logk denotes an iterated logarithm. A generalization of j and P above has been considered by J. M. De Koninck, I. Katai´ and A. Mercier [229]. Let now f1(n) = p j , where j−1 j α α i ≤ 1/2 < i ( ) pi n pi 37 i=1 i=1 α It is not difﬁcult to prove that the density of integers n for which f1(n)0 there is a Kε so that for every k > Kε the density of integers n for which k 1 −ε 1 +ε k 2 k 2 n < f1 (n + i) < n (39) i=1 is greater than 1 − ε?

~~335 CHAPTER 4~~

Another function introduced by P. Erdos¨ in [98] is α α α + ( ) = i , i ≤ < i 1 ( ) f2 n pi pi n pi 40 f (n) Then 2 →∞on a sequence of upper density 1. (41) n Erdos¨ conjectures that the logarithmic density L(α) of integers n satisfying f (n) 2 ≥ α(42) n exists, and is a continuous strictly decreasing function of α, L(0) = 1, L(∞) = 1. The ordinary density will not exist, and even the mean value of f2 will not exist on base of (41). Another conjecture is that x log x max f2(n) = (1 + o(1)) (43) n

4 Other functions; the derived sequence of a number r = a1 ... ar ( ) = k If n p1 pr , and Bk n ai pi , then many properties of Bk can be i=1 found in [260] (pp. 143-149). A similar function is r g(n) = 1 + ai (pi − 1), g(1) = 1,(44) i=1 introduced by Euler, and studied by H. W. Gould [149]. J. Sandor´ [299] introduced the function r k + ai pi 1 fk(n) = , fk(1) = 1 (45) i=1 2

~~Put f (n) = f1(n) in (45). Some simple inequalities satisﬁed by fk(n) are n f (ϕ(n)) ≤ (46) 2 with equality only for n = 1, 2, 3 (where ϕ (where ϕ is Euler’s totient), and~~

~~fk+m(n) ≥ fk(n) fm(n)(k, m ≥ 1)(47) σ ∗( ) Let k n be the sum of kth powers of unitary divisors of n. Then the normal order of magnitude of σ ∗(n) log k (48) fk(n) is log 2 log log n.~~

~~336 SPECIAL ARITHMETIC FUNCTIONS~~

~~One has also 1 σ ∗( )/ ( ) = lim sup log k n fk n n→∞ (n)~~

~~1 ∗ = lim inf log σ (n)/fk(n) = log 2 (49) n→∞ ω(n) k ∗ (n) Let D (n) = 2 . Then there exists a sequence (nk) such that~~

~~∗ log f (nk)D (nk)/nk log nk (50)~~

~~G. L. Cohen and D. E. Iannucci [56] deﬁne a multiplicative arithmetic function D such that D(pa) = apa−1 (p prime)(51)~~

− − β(n) and D(1) = 1. Since D(n) = a ...a pa1 1 ...par 1,wehaveD(n) = n, 1 r 1 r γ(n) where the function β appears in section 2 (see (17)), and γ(n) = p1 ...pr is the ”core” of n (see [260]), i.e. the greatest squarefree divisor of n. Let D(0)(n) = n and D(k)(n) = D(D(k−1)(n)) be the iteration of order k of D. The sequence

D(n) ={n, D(n), D(2)(n), ...} (52) is called the derived sequence of n.IfD j+k(n) = D j (n), where j ≥ 0 and k ≥ 1 is the smallest integer with this property, then D j (n), . ..,D j+k−1(n) is a derived k-cycle of n. The following result shows the existence of 5-cycles: Let s be such that (s, 2 · 3 · 5 · 47) = 1, and suppose (s1, 5 · 23 · 47) = 1, where s1 = (3s − 1)/2. Put 3s 45 5 (4) n = 2 · 3 · 5 · 23 · s1. Then n, D(n),...,D (n) is a 5-cycle. (53) Forsimilar results on 6 or 8-cycles, see [56]. Cohen and Iannucci prove also that for all n < 1.5 · 1010, the derived sequence D(n) is bounded; (54) but the density of integers n for which D(n) is unbounded, is less than 0.004 (55) It is not known the existence of unbounded derived sequences, but the authors conjecture this existence.

~~5 The consecutive prime divisors of a number~~

~~Let p1(n)~~

~~log log p j (n) = (1 + o(1)) j (56)~~

~~Forasimple proof of (56), see [100]. A more precise estimate is log log p j (n) = j + O( j log log j)(57)~~

337 CHAPTER 4 for almost all n,andall ξ(n) ≤ j ≤ ω(n), where ξ(n) →∞(see R. R. Hall and G. Tenenbaum [172]). Let λ j (p) = d{n : p j (n) = p} (58) be the density of integers n whose j-th prime factor is p.Bythe sieve of Eratosthenes it follows that 1 1 λ j (p) = 1 − 1/m (59) p q

0 w(t) = 1, t = 0 P. Erdos¨ and G. Tenenbaum [114] have proved that F(ρ, p)ρ1−k 1 λ (p) = 1 + O (60) j pF(1, p)w(k − 1) R uniformly, for all 1 ≤ j ≤ p1−ε (where ε>0isgiven), where ρ is the solution of the equation ρ = j − ( ) − + ρ 1 61 q

~~338 SPECIAL ARITHMETIC FUNCTIONS~~

~~Other corollaries are: 1 + O(1/ log log p) max λ j (p) = as p →∞ (64) j≥1 p 2π log log p and all values of j realizing this maximum satisfy~~

j = log log p + O(1), (65) + λ ( ) = − − − + 2 log2 j 1+ max j p exp j log j log2 j 1 p log j 2(log j)2 − log j + O(1) + 2 2 (66) (log j)2 = where log2 j log log j. Moreover, all values of p realizing this maximum satisfy j 2 log j 2(log j)2 − 3 log j + O(1) log p = 1 + 2 + 2 2 (67) log j log j (log j)2

~~Erdos¨ and Nicolas [110] have studied the functions~~

− − r 1 p (n) r 1 p (n) f (n) = i and F(n) = 1 − i + ( ) + ( ) i=1 pi 1 n i=1 pi 1 n They proved, for example, that there exists a constant c = 1.70 ... such that F(n) ≤ log n − c + o(1), for all n, and F(n) ≥ log n − c + o(1) for inﬁnitely many n. The proofs use besides results on the distribution of primes, also classical theorems of Optimization theory. The following results are due to P. Erdos¨ (see [101] and [100]). For almost all integers ∗ 1 1 = + o(1) log log log n (68) j 2 where ∗ indicates that log log p j (n)> j.Similarly, for almost all n, ∗∗ 1 1 = + o(1) log log log n (69) j 2 where ∗∗ indicates that p j (n)>p j (n + 1) in the summation. For almost all n one has ∗∗∗ 1 √ = (1 + o(1))c log log log n (70) j

339 CHAPTER 4 where ∗∗∗shows that the summation is extended over the j for which j < log log p j (n)< j + 1, and c is a constant. log log p (n) − j The function √j has normal distribution, and if j /j →∞, then j 1 2 log log p (n) − j log log p (n) − j √j1 1 and √j2 2 are asymptotical independent. (71) j1 j2 J. Galambos [137] deﬁned a number n to be weakly-composite if r 1 ≤ (r = ω(n)) ( ) ( ) 2 72 j=1 p j n He proved that (by solving a conjecture of I. Katai)´ for all sufﬁciently large integers n, there is a weakly composite number between

~~n and n + log log log n (73)~~

On the other hand, the behaviour of 1 = R (x)() ( ) j 74 n≤x p j n has been investigated by several authors for some specific values of j.Inparticular, for j = 1, or whenever j is preassigned, it is not difficult to prove that R j (x) ∼ c j x, with computable constants c j . When j = ω(n), the problem of finding good approximations to R j is more difficult. The case j = ω(n) − 1, or j = ω(n) − k with k fixed, shows no similarity to the above case. For asymptotic results (due to many authors, with refinements), see [260]. In order to obtain an average type of information on the magnitude of p j (n), when j is not fixed, or j = ω(n)−k,with k fixed, J. M. De Koninck and J. Galambos [230] have set up the following probabilistic approach. For every integer n ≤ x, pick one 1 p(n) ∈{p (n), . ..,p (n)} with equal probabilities , and consider the sums 1 r ω(n) 1 = R(x)() ( ) 75 n≤x p n Assume that x is an integer. Then there are ω(2)...ω(x) sums of the type (75) and Rr (x) ≤ R(x) ≤ R1(x). We shall say that a property holds for almost all sums in (75), if the number N(x) of the sums with the property in question satisfies

~~N(x)/ω(2)...ω(x) → 1asx →∞.~~

~~340 SPECIAL ARITHMETIC FUNCTIONS~~

De Koninck and J. Galambos [230] prove that for almost all sums in (75) one has cx x R(x) = + O ,(76) log log x (log log x)2 where c = 1/p2. The proof is based on a form of the Chebyshev inequality of pprime Probability theory. In another paper, De Koninck and Galambos [231] have studied the intermediate prime divisors of an integer. With density one, we can distinguish three types of prime divisors: we call p j ”small” if j is bounded as n →+∞; p j ”large” if ω(n)− j remains bounded as n →∞;all others are ”intermediate”. For the investigation of small prime divisors, tools from elementary number theory are sufﬁcient. Large prime divisors require special tools (due e.g. to Dickman, see e.g. [260] for asymptotic log p formulas involving large prime divisors), and old results show that j falls into log n the interval (a, b), where 0 ≤ a < b ≤ 1, with positive density, when j = ω(n). Extensions are known also for all large prime divisors, with similar results in nature. This perhaps explains why it was so important in probabilistic number the- log r ory to truncate additive functions at r = r(N) with → 0; it simply cancels log N the effect of the large prime divisors (see P. D. T. A. Elliott [95]). The truncation methods already show that the intermediate prime divisors behave asymptotically as independent random variables. In 1976 J. Galambos [138] showed that for intermediate terms, log log p j+1(n) − log log p j (n) are asymptotically unit exponential variables, (77) i.e. the density for which the above stated difference is smaller than 0 < z, equals 1−e−z.The remarkable part of this result is that the density does not depend on j.H. Maier [246] extended this result by showing that a ﬁnite set of the above differences are asymptotically independent in the sense of probability theory. De Koninck and Galambos [231] further generalize these results, by proving the following statement: Let j = j(N) be a positive integer valued function, j(N) →+∞as N → +∞.Assume that j is such that, with perhaps the exception of a set of density zero, p j (n) →+∞with N, and (log p j (n))/ log N → 0asN →+∞, where 1 ≤ n ≤ N. Then the points log log p j+k (k ≥ 1) form a Poisson process in limit as N →+∞. (78) In the proof, a result of A. Renyi´ [284] on the Poisson process is applied. For another notion of asymptotic prime divisor, see S. McAdam [1].

~~341 CHAPTER 4~~

~~6 The consecutive divisors of an integer~~

~~Let 1 = d1(n)~~

As in (75), De Koninck and Galambos [230] say that a property holds for almost all sums in (79) if it holds for M sums such that M/d(2)...d(x) → 1asx →∞ (where x is an integer, and d(k) denotes the number of distinct divisors of k). Then one has the following results (see [230]): For almost sums in (79), c x x Q(x) = 1 + O (80) log x (log x)3/2

~~Deﬁne similarly r(n) = S(x)(81) n≤x Then, for almost all sums in (81) one has c x2 x2 S(x) = 2 + O (82) log x (log x)3/2~~

Here c1, c2 are positive constants. We note that similar results can be deduced for the more general sums h(r(n)), where h is a general arithmetic function, n≤x satisfying certain conditions. In relation (58) we have deﬁned a density function on the consecutive prime divisors of an integer. Deﬁne similarly

~~ j (k) = d{n : d j (n) = k}, k ≥ 1.(83)~~

P. Erdos¨ and G. Tenenbaum [114] have shown that the following two assertions are equivalent: (i)j (k)>0; (ii) d(k) ≤ j ≤ k (84) Further, the following estimates are true: −γ + ( / ) + ( ) e O 1 log k 3 log3 k O 1 ≤ max j (k) ≤ (k →∞), (85) j k log k k log2 k

~~342 SPECIAL ARITHMETIC FUNCTIONS~~

~~γ where logm denotes an iterated logarithm, and is Euler’s constant. Moreover, for all j realizing this maximum,~~

~~d(k) ≤ j ≤ d(k)(log k)e log 2+o(1) (86) log j log j Similarly, for j ≥ 3, let K := exp 2 . Then log 2~~

~~−1 −(log j+O(1))/ log 2 −1 (log j+O(1))/ log 2 K j 3 ≤ max j (k) ≤ K j 3 (87) k as j →∞.Forall k realizing the maximum, one has~~

~~K α+o(1) ≤ k ≤ K 1+o(1) (88) where α = 0.293815 ... is the unique solution of the equation~~

~~(α + 1) log(α + 1) − α log α = log 2.~~

~~7 Functional limit theorems for the consecutive divisors~~

Let γx (...) denote the uniform probability measure on the set {1,...,[x]}; put Lu = log max{u, e}, and Lk = L(Lk−1). Then one has (see [172]) lim lim sup γ max |L d (m) − log k|≥(1 + ε) 2(log k)L k = 0 (89) →∞ x 2 k 2 2 3 n x→∞ n≤k≤d(m) which is the counterpart of the following result on prime divisors: lim lim sup γ max |L p (m) − k|≥(1 + ε) 2kL k = 0 (90) →∞ x 2 k 2 n x→∞ n≤k≤ω(m) These results were extended into the Strassen functional form by E. Manstaviciusˇ [249]. Let d(m, n) = card{d ∈ N : d|m, d ≤ u}. G. Tenenbaum [323] (see also [172]) showed that

t s γx (d(m, x ) − d(m, x )) < ud(m) ⇒ Fst(u), 0 ≤ s < t ≤ 1,(91) where ⇒ denotes weak convergence of distribution functions, Fst(u) is some purely discrete distribution function, and x →∞. This relation can be interpreted as referring to the increments of the arithmetic process d(m, xt ) Y = Y (m, t) = (92) x x d(m)

343 CHAPTER 4 having trajectories in the space D = D[0, 1] of real-valued functions on [0, 1] which are right-continuous and have left-hand limits. The behaviour of the expectation of this process is due to J. M. Deshouillers, F. Dress and G. Tenenbaum [75]: 1 2 √ Y (m, t) = t + o( ), ≤ t ≤ ( ) x π arcsin 1 0 1 93 x m≤x Let ω(m,v)= card{p : p|m, p ≤ v} (p = prime) and deﬁne the process

1 t ψx = ψx (m, t) := √ (ω(m, exp{(Lx) }) − tL2(x), 0 ≤ t ≤ 1 (94) L2x which ”simulates” the Brownian motion w = w(t) given on some probability space {,F, P}. Let ρ(·, ·) be the supremum metrics on the space D, and D be the Borel σ ρ γ ◦ ψ−1 D -algebra in D with respect to , and x x stand for the measure on deﬁned by γx (X x ∈ A), where A ∈ D, and 1 = ( , ) = √ ( ( , {( )t }) − ), ≤ ≤ ( ) X x X x m t log2 d m exp Lx tL2x 0 t 1 95 L2x Put µ for the Wiener measure P ◦ W −1.In1970 P. Billingsley [27] proved that the sequence of measures γ ◦ ψ−1 ( ) x x 96 weakly converges to the Wiener measure µ as x →∞(see also [28]). µ = γ ◦ −1 Denote x x X x . In 1996 E. Manstaviciusˇ [250] proved the similar result that the sequence of measures µx weakly converges to the Wiener measure µ as x →∞.(97) As a corollary, we note that (97) implies 1 u y2 γx (X x (m, t) − X x (m, s) 1. Then another corollary of (97) is: γ ( { ≤ ≤ ( , {( )t })> } < ) ⇒ ( ), ( ) x meas 0 t 1: log2 d m exp Lx tL2x u As u 99 where meas is the Lebesgue measure.

~~344 SPECIAL ARITHMETIC FUNCTIONS~~

~~In 1969 P. Erdos¨ [101] proved that~~

~~γx (card{k ≤ ω(m) : L2 pk(m)~~

+ and its counterpart for the divisors dk are stated in Manstaviciusˇ [250]: Let I denote the characteristic function of the set {y : y > 0}. Then we have γ 1 +( − ( )) <( ) ⇒ ( )() x I log2 k L2dk m L2 uL2x As u 101 k≤d(m) k Forasurvey of functional limit theorems in Probabilistic number theory, see the recent paper by E. Manstaviciusˇ [251].

8 Miscellaneous arithmetic functions connected with divisors In what follows we shall study various arithmetic functions connected with the divisors of a number. In 1973 P. Erdos¨ and S. K. Zaremba [118], in connection with work on good lattice points modulo composite numbers, have studied the arithmetical function

~~log d a(n) = (102) d|n d~~

~~Obviously, lim inf a(n) = 0. On the other hand, one has n→∞~~

a(n) lim sup = eγ ,(103) 2 n→∞ (log log n) where γ is Euler’s constant. The arithmetical function a(n) has a continuous purely singular distribution function. Let d+(n) denote the number of integers k, such that the interval [2k, 2k+1) contains at least a divisor d of n.(104) This function appears in Erdos¨ [100]. In [115] Erdos¨ and Tenenbaum proved that for all ε>0 there is c(ε) > 0such that for all 0 ≤ α ≤ 1, the upper density of all integers n satisfying d+(n) ≤ αd(n)(105) is less than c(ε)α1−ε. This implies a conjecture of Erdos,¨ namely that apart from a sequence of integers of density zero, d+(n) → 0asn →∞ (106) d(n)

~~345 CHAPTER 4~~

~~Various arithmetic functions have been studied by P. Edos¨ and J. L. Nicolas in [107], [108] and [109]. Let dn be the smallest divisor of n such that d ≥ n~~

~~d|n,d≤dn and put n f1(n) = (107) dn~~

~~In [107] one can ﬁnd that for ε>0 and n > n0(ε), one has~~

~~log2 n log3 n f1(n) ≤ exp(1 + ε) ,(108) log4 n and for inﬁnitely many m one has~~

~~log2 m log3 m f1(m) ≥ exp(1 − ε) (109) log4 m Another function introduced in [107] is 1 f2(n) = max d (110) 1≤k≤n k d|n,d≤k~~

~~It is proved that cd(n) log log n ≤ f2(n) ≤ c d(n) ,(111) log n log log n log n with c, c > 0 constants. The function f2(n) is supermultiplicative, i.e.~~

~~(m, n) = 1 ⇒ f2(mn) ≥ f2(m) f2(n)(112)~~

~~Let k0 = k0(n) the value of k in (110) for which the maximum is realized, i.e. 1 f2(n) = d. k0(n) d|n,d≤k0(n)~~

~~In [109], by probabilistic arguments it is shown that for all constants c1 ≤ 0.23 and sufﬁciently large n,wehave~~

~~k0(n) ≥ exp(c1 log n/ log log n); (113)~~

~~346 SPECIAL ARITHMETIC FUNCTIONS~~

~~and that there exists a constant c2 > 0such that for inﬁnitely many n,~~

~~k0(n) ≤ exp(c2 log n/ log log n)(114)~~

~~Another result says that for almost all n one has~~

~~k0(n)~~

The function f2 is connected with another function F deﬁned by F(n) = max 1 (116) t | , t < ≤ d n 2 d t via the relations 1 F(n) ≤ f (n) ≤ 2F(n), (117) 2 2 and this is exploited in [108]. A number n is called F-highly abundant if m < n ⇒ F(m) 0 such that d(n) d(n) c1 ≤ F(n) ≤ c2 (118) log n log log n log n log log n ρ ( ) = In 1976 G. Tenenbaum√ [324] introduced the functions 1 n greatest divisor of ρ ( ) = n less√ than or equal to n;and 2 n smallest divisor of n greatest than or equal to n.(119) He showed that for all ε>0 there is an absolute constant c and an x0(ε) such that 3/2 −α−ε 3/2 −α −1/2 x (log x) < ρ1(n)

~~(ρ1(n), ρ2(n)) = 1 (122) is equal to 1. Certain arithmetic functions related to distinct divisors have been studied by P. Erdos¨ and C. Pomerance [111] and [112].~~

~~347 CHAPTER 4~~

Let f (n) denote the least integer so that in the interval (n, f (n)] there are distinct integers a1,...,an with i|ai for i = 1, 2,...,n.For example, f (10) = 24, since a1 = 11, a2 = 22, a3 = 21, a4 = 16, a5 = 15, a6 = 12, a7 = 14, a8 = 24, a9 = 18, a10 = 20. More generally, if m is any positive integer, let f (m, n) denote the least integer so that in (m, m + f (n, m)] there are distinct integers a1,...,an with i|ai ,for i = 1, 2,...,n.(123) In [111] it is proved that for n ≥ 3, 2 √ + o(1) n log n/ log log n ≤ f (n) ≤ (2 + o(1))n log n (124) e The right side of (124) holds true also for n = 2. For f (n, m) the following upper bound is valid: √ f (n, m) ≤ 4n([ n] + 1), m, n ≥ 1 (125)

The upper bounds in (124) and (125) rely on a theorem of D. Konig¨ [225] and P. Hall [167] (called also as the ”Marriage theorem”) on bipartite graphs. Let L(n) = [1, 2,...,n]betheleast common multiple of 1, 2,...,n. Then for all sufﬁciently large n,

~~( ) 1 L n f (n, m)>n( n)α ( ) ( ) log 126 L n m=1 where α = .08607 ... is the constant of (120); and for all n~~

~~( ) 1 L n f (n, m) ≤ n ((β + o( )) n/ n), ( ) ( ) exp 1 log log log 127 L n m=1 where β = log(55/2/2 · 33/2) = 1.6825 ... The authors conjecture that~~

~~f (n, m) ≤ n1+o(1), and that~~

( ) (128) 1 L n f (n, m) n( n)γ γ> ( ) log for some 0 L n m=1 Related to Grimm’s conjecture, in [112] the following concepts are introduced. We say that a set of positive integers S has the distinct divisor property if for each s ∈ S we can ﬁnd a divisor ds|s,1≤ ds < s such that the ds are all distinct. If c > 0, let f (n, c) denote the cardinality of the largest subset of [n, cn] which has the distinct divisor property.

~~348 SPECIAL ARITHMETIC FUNCTIONS~~

~~By applying again the Marriage theorem, in [112] it is proved that for each c > 1 there is a constant δ(c) such that~~

~~f (n; c) ∼ δ(c)n as n →∞.(129)~~

Further, δ is a continuous and strictly increasing function, and δ(c) δ(c) 1 1 δ(c) lim = 1, lim = , < < 1 for all c > 1 (130) c→1+ c − 1 c→∞ c − 1 2 2 c − 1 δ(c) The following Open Problem is stated: is a monotonic function? c − 1

~~9 Arithmetic functions of consecutive divisors~~

Let 1 = d1 < d2 < ··· < dk(n)(k = d(n)) be the consecutive divisors of n.Inwhat follows we shall study certain arithmetic functions connected to these consecutive divisors. In 1948 P. Erdos¨ [102] proved that the density of all integers n having two divisors d and d such that d < d ≤ 2d is positive. (131) He conjectured that this density is 1, or written equivalently d + E(n) = min i 1 :1≤ i ≤ k → 1a.e. (132) di (where a.e. - almost everywhere - means that the relation holds true on a sequence of density one). Conjecture (132) has been established in 1984 by H. Maier and G. Tenenbaum [247]. The arithmetical function

f (n) = card{1 ≤ i ≤ k : (di , di+1) = 1} (133) has been introduced by P. Erdos¨ and R. R. Hall in [106], while its ”dual” is the Erdos-¨ Montgomery function from 1979 (see [116]), deﬁned by

~~g(n) = card{1 ≤ i ≤ k : di |di+1} (134)~~

~~A more general function than f is fr , introduced in [106], too, is ~~

~~fr (n) = card 1 ≤ i ≤ k − r + 1: max (d j , dk) = 1 (135) 1≤ j~~

~~Then f2(n) = f (n).~~

~~349 CHAPTER 4~~

By improving earlier results, Erdos¨ and Tenenbaum [117] prove the following results: 3 log log(2d(n)) f (n) ≤ (n ≥ 2), (136) log d(n) f (n) ≤ 4(log(1 + (n))/(n))1/3 (n ≥ 2)(137) d(n) 5 f (n) d(n)1−c, c = − log 3/ log 2 = 0.0817 ... (138) 3 On the other hand, f (n) x(log x)α (139) n≤x where α = 3 · 4−1/3 − 1 = 0.8898 ...; and f (n) x(log x)log 3−1+o(1), x ≥ 3 (140) n≤x

~~In fact, (140) is a corollary of a result on an arithmetic function h(n), introduced by Erdos¨ and Tenenbaum ([117]) as~~

~~k ∗ d h(n) = i ,(141) + i=1 di 1 where ∗ indicates that in the sum one takes (di , di+1) = 1. The result on h(n) is the following: h(n) = (log n)log 3−1+o(1) a.e. (142)~~

~~The following theorem improves essentially an earlier estimate from [106]:~~

~~max f (n) ≥ exp{((log 2)2 + o(1)) log x/(log log x)2} (143) n≤x~~

~~The similar result for fr (n) is 1 2 max fr (n) ≥ exp + o(1) (log log x) / log log log x (144) n≤x 2~~

~~In [325] G. Tenenbaum introduced the function 1 d + ψ(n) = max log i 1 :1≤ i ≤ k − 1 (145) log n di~~

350 SPECIAL ARITHMETIC FUNCTIONS and proved that is has a continuous distribution function d(x) on [0, 1]. This functions satisﬁes a1x ≤ d(x) ≤ a2x log(2/x) (a1, a2 -constants), see [327]. Another function, by Hall and Tenenbaum (see [115]) introduced in 1979 is d α U(n,α)= card d|n, d |n : (d, d ) = 1, log ≤ (log n) (146) d In [172] it is proved that

U(n,α)∼ 1 + (log n)log 3−1+α+o(1) a.e.,(147) which for α ≤ 0 contains the Erdos¨ conjecture (132), and which is used also in the proof of (142). Other properties of U(n,α)have been obtained by R. R. Hall [168]: U(n,α)/(2α + 2)ω(n) = C · x + O(x(log x)−γ ), n≤x = (α), γ = γ(α)> ; where C C 0 ( , )/ ω(n) ; U n 0 2 x log log x (147) n≤x 3 5 U(n, 1)/4ω(n) ∼ · C · x n≤x 4 4 as x →∞ Erdos¨ and Hall [106] considered the important function

~~k(n) = card{d : d(d + 1)|n} (148) and showed that √ max k(n)>(log x) e+o(1) (x →∞), (149) n≤x and which was strengthened to~~

/( ) max k(n)>(log x)log3 x 9 log4 x (x →∞)(150) n≤x by A. Balog, O. Erdos¨ and G. Tenenbaum [15]. A generalization of k is the function ks introduced by R. de la Breteche` [39], [40]. Put ks(n) = card{d : (d, s) = 1, d(d + s)|n} (151) log n/ log log n Clearly k1 ≡ k. Let D(n) = 2 . Then (see [40]):

~~c1+o(1) ks(n) ≤ D(n) , as n →∞ (152) uniformly for all s ≥ 1, with c1 = 0.565 ...~~

~~351 CHAPTER 4~~

~~In 1986 J. M. De Koninck and A. Ivic[´ 232] have studied two functions, namely~~

− k 1 1 H1(n) = , k = d(n), (153) + − i=1 di 1 di and r−1 ( ) = 1 , = ω( ), ( ) H1 n r n 154 + − i=1 pi 1 pi where p1 < p2 < ···< pr are the consecutive prime divisors of n. They proved that x log log x H (n) = Ax + O (155) 1 log x where A = 0.299 ... is a constant; and −1/3 H1(n) = B · x + O(x(log x) )(156) n≤x with another constant B = 1.77 ... In [117] Erdos¨ and Tenenbaum improved the error term in (156) to

~~O(x(log x)−1(log log x)3)(157)~~

~~They proved also that~~

~~− 1 +ε H1(n) d(n)(log d(n)) 3 (n ≥ 2)(158) for any ﬁxed ε>0, and that~~

1 +o(1) max H1(n)>exp{(log x) 2 } (x →∞)(159) n≤x Further, 1−c2 H1(n) d(n) log(1 + ω(n)) (160) 5 log 3 whenever n is squarefree, where c = − = 0.08170 ... 2 3 log 2 In [15] it is proved that 1−c2+o(1) max H1(n) ≤ max d(n) (x →∞), (161) n≤x n≤x where c2 is the same as in (160). It is conjectured that a bound of type

~~1−δ H1(n) d(n) (162) with an absolute δ>0 holds unconditionally.~~

~~352 SPECIAL ARITHMETIC FUNCTIONS~~

~~The function H1 has a distribution function (see [117]), and this is everywhere continuous on the real line (see [15]). In [116] the authors study also a general function of type~~

− k 1 d F(n,θ)= θ i ,(163) + i=1 di 1 where θ :[0, 1] → R is an arbitrary function. They show that when θ is bounded, then the function F(n,θ) (164) d(n) has a distribution function. When θ is of class C2, then an asymptotic formula of type log log log x F(n,θ)= x log x θ(1) + O (165) ( )δ n≤x log x log log x is valid, where δ = 0.086 ... is an optimal constant, if one assumes that tθ (t) is a monotonic function. For the particular case θ(t) = tr a more precise estimation is true: r F(n, t → t ) = x(log x − Kr (x) + O(1)) (166) n≤x uniformly for r log x 1, where Kr (x) satisfy the double inequality ( ) −εr Kr x δ e L (log x) ε (1 + r )L ((r/(r + 1)) log x), (167) 1 (r log x)1−δ 2 with L1(v) = exp{−c(ε) log v log log v}, −1/2 L2(v) = (log v) log log v +ε 1 1 In the case of θ(t) = − 1 (ε > 0),M.D.Vose [345] constructed a t sequence (Nn)n≥1 such that F(Nn,θ)= Oε(1), solving a conjecture by Erdos.¨ In fact, the above relation holds true also by replacing Nn with the sequence of factorials (n!) or with lcm[1, 2,...,n], see G. Tenenbaum [334], who studied the 1 more general case of θ(t) = h − 1 , where h :[0, ∞) → [0, ∞) belongs to a t certain class generalizing the one considered by Vose.

~~353 CHAPTER 4~~

For similar or related problems on the sequence of factorials, as well as the distribution of divisors of factorials, see M. D. Vose [346] and D. Berend and J. E. Harmse [21]. Another function of type (153) is Tenenbaum’s function ([326]) 1 H (n) = ( ) 2 − 168 1≤i< j≤k d j di

Tenenbaum proves that 1−c H2(n) d(n) (169) with an explicit positive constant c.Hefurther shows that the inequality ξ ξ k i j ≤ M ξ 2 (ξ ∈ R), ( ) − i i 170 1≤i< j≤k d j di i=1 which is a special case of Hilbert’s inequality, holds with a constant M = M(n) satisfying M(n) log log d(n) as n →∞and M(ni ) log log d(ni ) for a suitable sequence (ni ) with d(ni ) →∞as i →∞. R. de la Breteche` [39] shows that

~~c3 H2(n) d(n) log d(n) log log(2d(n)), (171) where c3 = 0.918 ... is a constant; and in [40] that~~

c1+o(1) H2(n) D(n) ,(172) where D(n) and c1 are as in (152). In [326] Tenenbaum has introduced a general class of arithmetical functions F. We say that an application gn, deﬁned on the set of divisors of n with integer values, is regular if gn is injective, and for all divisors d, t of n one has

~~((d, gn(d)) = 1,(t, gn(t)) = 1, dgn(d) = tgn(t)|n) ⇒ d = t (173)~~

~~Let F be the class of arithmetic functions F ∈ F deﬁned by~~

~~F(n) = card{ f |n : gn(d)|n,(d, gn(d)) = 1} (174)~~

~~Let fF (n) = max{F(n) : F ∈ F} (175) This function is a common generalization of the functions f and k deﬁned in (133) and (151), respectively.~~

~~354 SPECIAL ARITHMETIC FUNCTIONS~~

~~The properties of fF have been recently investigated in [39], where the following are proved: fF is super-multiplicative, i.e.~~

~~fF (mn) ≥ fF (m) fF (n) for all (m, n) = 1; (176)~~

~~c3 fF (n) ≤ d(n) for all n,(177) where c3 is the constant of (171);~~

~~log fF (n) lim sup = c3; (178) d(n)→∞ log d(n)~~

d(n)1/2 fF (n) ≥ max(min{d(k), d(n/k)}) ≥ √ ; (179) k n H(n) + 1 where k n means that k is a unitary divisor of n (i.e. k|n, (k, n/k) =1), and H(n) is the maximum of the exponents γ in the prime factorization n = pγ (i.e. the pγ n function deﬁned by (1) and studied in section 1). One has also √ 2 γ f (n) ≤ d(n) ( ) F γ + 180 pγ n 1

~~As a corollary, one obtains d(n) f (n) = O (181) F (n)~~

Let p(n) denote the least prime divisor of n.Bystudying certain sieve- theoretical problems, in 1959 A. Schinzel and G. Szekeres [290] considered the function S(n) = max{dp(d) : d > 1, d|n} (n ≥ 2); S(1) = 1 (182) This function is connected with the set of consecutive divisors by the surprizing formula S(n) d + = max i 1 :1≤ i ≤ k − 1 = E∗(n), (183) n di due to G. Tenenbaum [327]. Let D(x, y), A(x, y) be distribution functions deﬁned by

~~D(x, y) = card{n ≤ x : S(n) ≤ yn}, (184) A(x, y) = card{n ≤ x : S(n) ≤ yx}~~

355 CHAPTER 4 where x ≥ 1, y > 0. By (183) one can say that D(x, y) is exactly the distribution ∗ function of E (n).ForD and E the following estimates are true (see [327]): δ ε, λ ε> / λ>5 − log = . ... Let real numbers such that 5 3, 1 δ 4 20 √ 3 1 + 5 δ = log .Thenfor all x ≥ y ≥ 2 one has 2

~~x x L(x, y) ,λ D(x, y) ≤ A(x, y) log 2u,(185) u y u log x where u = ,and log y (log u)−λ, 2 ≤ y ≤ exp{(log log x)ε} L(u, y) = 1, y > exp{(log log x)ε}~~

~~The following inequality for the Schinzel-Szekeres function is true:~~

~~S(mn) ≤ max{S(m) · n, S(n)} (m, n ≥ 1)(186) with equality if P(m) ≤ p(n) (where P(m) denotes the greatest prime factor of m). As a corollary,~~

~~S(mp) ≤ pS(m) for all m ≥ 2, p ≤ S(m)(p = prime), (187) which follows by (186), since S(p) = p2 for a prime p. In analogy with (184) deﬁne~~

~~D(x, y, z) = card{n ≤ x : S(n) ≤ ny, p(n)>z} (188) A(x, y, z) = card{n ≤ x : S(n) ≤ xy, p(n)>z}~~

~~E. Saias [290] found that there exist three positive constants c4, c5 and y0 such that under the conditions~~

~~0.535 5 x ≥ y ≥ z + z ≥ 3, x ≥ z and y ≥ y0 (189) we have x log(y/z) x log(y/z) c · ≤ D(x, y, z) ≤ c · (190) 4 log x log z 5 log x log z log x log x Let u = and v = . log y log z~~

~~356 SPECIAL ARITHMETIC FUNCTIONS~~

For v>1, the Buchstab function ω(v) is defined as the unique continuous solution to the difference-differential equation (vω(v)) = ω(v − 1)(v>2) (191) vω(v) = 1 (1 ≤ v ≤ 2) Let ω(v) = 0 for v<1, and define ω at 1 and ω at 1 and 2 by right continuity. The Buchstab function arises in the study of D(x, x, z).For an overview of results on these functions, see [260] (Chapter 4). For u > 1wedefine the function d(u,v)as the unique continuous solution to the equation ∂ u(v − 1) 2u (vd(u,v))= d ,v− 1 v>u > 1,v> (192) ∂v u + v u − 1 with initial conditions   ω(v), ≤ < ,  if 0 u 1 ( ,v)= ω(v) − u ω( ), ≤

357 CHAPTER 4 where v 2 Wε = (u,v)∈ R :30isgiven). In [328] G. Tenenbaum showed that for λ ∈ [0, 1] ﬁxed, S(n) card n ≤ x : ≤ nλ ∼ xs(λ) (x →∞), (196) n where s is a continuous function. Put

~~d(u) = s(1/u)~~

~~Then for u ≥ 1ﬁxed~~

~~D(x, y) := D(x, y, 1) ∼ xd(u) as x →∞ (197) A(x, y) := A(x, y, 1) ∼ xd(u)~~

~~In 1997 E. Saias [291] showed that there exist constants a1 and a2 such that~~

~~a1 ≤ ud(u) ≤ a2 for u ≥ 1 (198)~~

~~In [349] it is shown that for 1 ≤ u ≤ v,~~

~~d(u,v)= e−γ d(u) + O(1/v) (199) and using (194), (199) it follows that~~

1 · 8 3 · 4 ≤ d(u) ≤ , u ≥ 1 (200) u + 1 u + 1 which is a precision of (198). Similarly, if u ≥ 1ﬁxed, and v →∞, from the above one gets 1 D(x, y, z) ∼ D(x, y) 1 − (x →∞) p≤z p (201) 1 A(x, y, z) ∼ A(x, y) 1 − p≤z p

~~358 SPECIAL ARITHMETIC FUNCTIONS~~

Explicit formulas for d(u) and d(u,v)are given in [291], resp. [349]. These are: 1 − yk 1 d(u) = ρ dx1dx2 ...dxk (202) xk x1x2 ...xk A 1 where A = 0 < xk ≤ xk−1 ≤ ···≤ x1 ≤ 1, max (x j + y j ) ≤ 1 + , and ρ is the 1≤ j≤k u Dickman function, deﬁned to be the unique continuous solution to the equation

~~vρ (v) + ρ(v − 1) = 0 (v > 1) ρ(v) = 1 (0 ≤ v ≤ 1), ρ(v) = 0 (v < 0), ρ(v) (203) extended to v = 1byright continuity~~

For properties of ρ,see [260] (Chapter 4). Here y j = x1 + x2 + ···+x j and k = [u]. Similarly, 1 − yk 1 − yk 1 d(u,v)= σ , dx1 ...dxk (204) xk 1/v x1 ...xk A where A, k and y j are as above, and σ(u,v) is the unique solution, continuous for u > 1, to ∂σ v u (u,v)+ σ(u − 1,v− v/u) = 0 u > ,v>1 ∂u v − 1 with initial conditions σ(u,v)= ω(v) (0 ≤ u < 1), (205) u σ(u,v)= ω(v) − 1/v 1 ≤ u ≤ v ≤ . u − 1

This function has been studied by J. B. Friedlander [135], and E. Saias [292]. We note that results on D(x, y) and A(x, y) of (184) have been applied to the small sieve of Erdos¨ and Ruzsa ([289]) and to practical numbers e.g. in [327] or [293]. E. Saias [291] improved result (185) to x x c ≤ D(x, y) ≤ A(x, y) ≤ c (x ≥ y ≥ 2), (206) 6 u 7 u where c6, c7 > 0 are constants.

~~359 CHAPTER 4~~

In a recent paper [350] A. Weingartner proves that for any ε>0, for x ≥ 3 and x ≥ y ≥ exp{(log log x)5/3+ε},wehave x 1 A(x, y) = xd(u) + O = xd(u) 1 + O (207) log x log y and a similar result for D(x, y). Let B(x, y, z) be deﬁned by

~~B(x, y, z) = card{n ≤ x : S(n) ≤ xz, P(n) ≤ y} (208)~~

Let a function a(u,v) deﬁned by a(u,v) = 0ifu < 0orv<0, = ρ(u) if u >v≥ 0, = ρ(v) + A(u,v)for 0 ≤ u ≤ v, where ρ is the Dickman function, and v ds A(u,v)= a(s − 1,v(1 − 1/s)) . , 2v s max u v+1

Then in [350] it is proved that for x ≥ 3, and x ≥ y ≥ z ≥ exp{(log log x)5/3+ε} we have 1 log u B(x, y, z) = xa(u,v) 1 + O + (209) log z log y A similar result holds true when B(x, y, z) is replaced with

~~E(x, y, z) = card{n ≤ x : S(n) ≤ nz, P(n) ≤ y} (210)~~

10 Hooley’s function An important arithmetic function (with applications to many parts of Number theory) is the function by C. Hooley, introduced in 1979 [185]. For n ≥ 1 and all real x put (n, x) = card{d|n : x < log d ≤ x + 1} and ( ) (n) = max (n, x) 211 x∈R This function satisﬁes the elementary inequalities

~~1 ≤ (n) ≤ d(n)(n ≥ 1); (212) (mn) ≤ (m)d(n)(m, n ≥ 1)~~

~~Put S(x) = (n). n≤x Hooley proved that 4 − S(x) x(log x) π 1 (213)~~

~~360 SPECIAL ARITHMETIC FUNCTIONS and that S(x) →∞as x →∞ (214) x Hooley in [185] introduced also the more general function~~

~~r (n) = max card{d1,...,dr−1 : d1 ...dr−1|n, x1,...,xr−1~~

xi < log di ≤ xi+1, 1 ≤ i < r} (215) and with Sr (x) = r (n) n≤x showed that √ r−1 Sr (x) r x(log x) (r ≥ 2)(216) 4 − = . ... In 1982 Hall and Tenenbaum [173] improved the exponent π 1 0 27323 in (213) to 0.23457 ...,and with an interesting more elaborate method, to 0.23454. Furthermore, they proved that

~~S(x) x log x (217) and (n)<(log n)B for almost all n,(218) log 2 where B = log 2 − + ε. log 3 In 1984 and 1985 Maier and Tenenbaum [247], [248] have obtained the strong improvements~~

~~(log log n)γ <(n)<ξ(n) log log n for almost all n,(219) for all constants γ<0.28754 ..., and all functions ξ(n) →∞as n →∞. The left side of (219) implies~~

(n)>1 for almost all n,(220) and this contains essentially the famous conjecture of Erdos¨ (see also (132)). In [172] it is proved that √ 2+ε S(x) ε xL(log x) (221) where L(u) = exp{ log u log log u}, (u ≥ 3), while for Sr (x) Maier and Tenenbaum deduced (related to (216)),

~~αr +ε x log log x Sr (x) ε,r xL(log x) (221 )~~

~~361 CHAPTER 4~~

~~for all r ≥ 2, where the constants αr satisfy " 1 α ≤ (r − 1) r(r + 2)(r ≥ 2)(222) r 2~~

~~Forcertain applications, it is important to introduce also the pondered sums ω(n) Sr (x, y) = r (n)y (223) n≤x~~

In 1985, Hall and Tenenbaum (see [172]) showed that Cr log r S (x, y) x(log x)y−1 exp (log x2) for 0 < y < 1,(234) r r 1 − y 3 and r(y−1) αr +ε Sr (x, y) r,y,ε x(log x) L(log x) for y ≥ 1,ε>0,(235) where the constants αr satisfy (222). In 1986 they obtained the asymptotic majorization √ t ω(n) β(t,y)−1 2 t+o(1) (n) y t,y x(log x) L(log x) (236) n≤x for all t, y satisfying t ≥ 1, y ≥ t/(2t − 1), where β(t, y) = 2t y − t, and L is deﬁned as above. In 1990 G. Tenenbaum [329] proved a partial generalization of (236): Let F(X) be an irreducible polynomial over Z[X]ofdegree > 1. Then for all t ≥ 1 one has √ t β(t)−1 2t+o(1) (F(n)) t x(log x) L(log x) , x →∞ (237) n≤x where β(t) = 2t − t. Let M(x) denote the number of integers n ≤ x such that

~~(n) = 1.~~

~~A quantitative version of (n) = o(n) is the following (see [330]):~~

~~−β+ε M(x) ε x(log log x) ,(238) where β = 1 − (1 + log log 3)/ log 3 = 0.00415 ...~~

~~362 SPECIAL ARITHMETIC FUNCTIONS~~

~~11 Extensions of the Erdos¨ conjecture (theorem) Let~~

~~ −λ Bλ(n) ={m : there exists d|n and d |m such that d < d ≤ (1 + (log n) )d}, where λ ≥ 0. Then the Erdos¨ conjecture can be written as~~

dB0(n) = 1 + o(1) for almost all n (239) where d denotes asymptotic density. In 1995 A. Rouj [283] proved a generalization of (239) in a strong effective form. In part I he proved that √ −cλ log log n −Q(β)+o(1) 1 − e ≤ dBλ(n) ≤ 1 − (log n) (240) for almost all n,whereβ = ((1 + λ)/ log 2) − 1 and Q(β) = β log β − β + 1. In part II, it is shown that for almost all n we have

−F(λ)+o(1) dBλ(n) = (log n) ,(241) where F(λ) = 0 for λ ≤ log 4 − 1, F(λ) = Q(β) for log 4 − 1 <λ≤ log 8 − 1, F(λ) = λ − log 2 for λ>log 8 − 1. An analogue of the Erdos¨ conjecture result is due to M. Car [51]. Let Fq [X] denote the ﬁnite ﬁeld of order q. Then almost all (monic) polynomials of degree n in Fq [X] possess two distinct divisors of the same degree. (242) Let t(n) be the number of exceptional polynomials. Then for any ε>0,

~~n −b+ε t(n) = Oε(q (log n) ), (243) 1 where b = (1 − ((1 + log log 3)/ log 3) = 0.00103 ... 4~~

12 The divisors in residue classes and in intervals We now state certain results on the distribution of divisors in residue classes and in intervals. Let k and l be integers satisfying 0 < l < k, (l, k) = 1. Denote by f (x, k, l) the number of integers n < x which have a divisor t satisfying t ≡ l (mod k); and F(x, k) denote the number of integers n < x which have a divisor ≡ l (mod k) for every l. Clearly F(x, k) ≤ f (x, k, l).Itiseasy to see that for fixed k, F(x, k) = x + o(x).Ifx tends to infinity with k, the questions become much difficult. In 1964

~~363 CHAPTER 4~~

Erdos¨ [103] proved that for any fixed ε>0 and k < 2(1−ε) log log x ,wehave, uniformly in k, F(x, k) = x + o(x)(244) In other words, if k < 2(1−ε) log log x , then almost all numbers have a divisor in every residue class l (mod k). Similarly, let k > 2(1+ε) log log x . Then uniformly in k and l, x f (x, k, l) = + o(x)(245) l 1 (1−ε) log log x Finally, let k < 2 2 and denote by d(n, k, l) the number of divisors of n which are ≡ l (mod k). Then for every η>0, we have for every l1 and l2,forall but o(x) integers n < x, d(n, k, l ) 1 − η< 1 < 1 + η(246) d(n, k, l2) Erdos¨ and√ Hall [169] proved that for any fixed real c,byassuming that k = 2log log x+(c+o(1)) log log x , one has x ∞ 1 F(x, k) ∼ exp − y2 dy as x →∞ (247) 2π c 2 Let A(k,α)denote the density of integers having a divisor t ≡ 1 (mod k) in the α range 1 < t < exp(k ).In[100] Erdos¨ conjectured the existence of α0 > 1with the property that A(k,α) → 0or1ask →∞, depending on whether the fixed value of α satisfies α<α0 or α>α0.In1992 R. R. Hall [169] prove this conjecture with α0 = 1/ log 2, and makes further progress by allowing α to depend on k, and 1 letting α → α ,ask →∞.For example, if α< − c log k/ log k, where 0 log 2 3 c > 2/ log 2, then A(k,α)→ 0ask →∞ (248) ∗ Let A (k,α)be the density of integers n such that for every (l, k) = 1, the number n has a divisor t ≡ l (mod k) in the interval exp exp( log k)

~~A∗(k,α)→ 1ask →∞ (249)~~

The proofs use the law of iterated logarithms of Probability theory, as well as results from Probabilistic group theory. Let H(x, y, z) denote the number of integers n ≤ x, which have at least one divisor d satisfying y < d ≤ z.The ﬁrst results on H(x, y, z) are due to S. Besicovitch,

364 SPECIAL ARITHMETIC FUNCTIONS these were sharpened and extended by Erdos¨ in 1952 and Tenenbaum in 1981 (see e.g. [329] and the References therein). Their result is: 1 lim H(x, y, 2y) = (log y)−δ+o(1),(y ≤ x1−ε), (250) x→∞ x where δ = 1 − log(e log 2)/ log 2 = 0.08607 ... and ε>0isfixed. More generally, let F(x) be an irreducible polynomial with integer coefficients and of degree > 1. If HF (x, y, z) is the number of n ≤ x such that F(n) has at least a divisor d such that y < d ≤ z, then −δ+o(1) HF (x, y, 2y) = x(log y) (251) when x, y →∞and y ≤ x1−ε. In [329] it is shown also that for any η>log 4 − 1, −η HF (x, y, 2y)>x(log x) (252) 1 when x, y →∞in the domain y ≤ x. 2 If 1 < y ≤ z < x, then in [331] it is proved that uniformly in x, y, z we have H(x, y, z) = x(1 + O(log y/ log z)) (253) ε Further, for every ε>0andy z < x,fory > y0(ε) we have x − H(x, y, z) εx(log y/ log z)(254) √ When 3 ≤ y ≤ z ≤ x,in1984 Tenenbaum proved (see [332]) that if z = y + y(log y)−β with a fixed β ≥ 0, then H(x, y, z) = x(log x)−G(β)+o(1),(255) where   + β + β 1 1 − + , β< − (β) = log 1 1 if log 4 1 G  log 2 log 2 β, if β ≥ log 4 − 1. When β = log 4 − 1 + ξ/ log log y, Hall and Tenenbaum [174] were able to show that H(x, y, z) x(log y)−G(β)(1 + max{−ξ,0})−1 (256) uniformly in ξ ≥−C(log log y)1/6 for any positive constant C. When y = eu, z = eu+1 one has a connection with Hooley’s function ;inthis case one has (see [332]): 1 Uniformly for 3 ≤ u ≤ log u, 2 H(x, y, z) = H(x, eu, eu+1) = xu−δ exp(O log u log log u)(257) where δ = 0.08607 ...

~~365 CHAPTER 4~~

13 Divisor density and distribution (mod 1) on divisors We shall consider the notions of divisor density of a sequence, due to R. R. Hall [170], [171], as well as that of distribution (mod 1) on divisors, due to I. Katai´ [205]. Asequence A has divisor density DA = z,ifthere is a sequence of integers S of asymptotic density 1 such that for d(n, A) = card{d|n : d ∈ A} one has

d(n, A) ∼ zd(n) as n →∞for n ∈ S (258) (n) Let gk(x, A) = (χA(n)−z)/4 ·n, where χA is the characteristic function n0andall0< z < 1; i.e. the function f (x) = (log x)α is equidistributed (mod 1) with respect to divisors, conjectured by Hall. In fact, it was Katai´ who ﬁrst studied equidistribution on divisors of an additive arithmetic function satisfying for all 0 ≤ z ≤ 1,

~~card{d|n : f (d) ≤ z (mod 1)}=(z + o(1))d(n).~~

Katai´ proved that an additive function f satisﬁes this property iff ν f (p) 2/p =+∞ (ν = 1, 2,...) (260) pprime where x denotes the distance of x to the nearest integer. Foranarbitrary arithmetic function f , (259) has the following consequence: e(ν f (n)) = o( x)(ν= , ,...) ( ) (n) log 1 2 261 k1, see [170], for f (x) = θx α (θ ∈ R \ Q), see [94]. Another function, f (x) = x for α ∈ R+ \ N was studied by Hall and Tenenbaum.

~~366 SPECIAL ARITHMETIC FUNCTIONS~~

14 The fractal structure of divisors When d runs through the divisors of a typical integer n, the numbers log d/ log n exhibit a fractal-like behaviour on the unit interval. This has been studied in 1993 by M. Mendes` France and G. Tenenbaum [258]. More generally, consider a dynamical system of points X = (Xn)n≥1, where for each n,

~~( ) ( ) ( ) X ={0 = x n < x n < ···< x n = 1} with k →∞as n →∞. n 0 1 kn n~~

Let ε = (εn)n≥1 such that εn → 0asn →∞, and put log(1/ε ) s(ε) = s(ε, X) = lim sup n n→∞ log kn ε(α) = ε(α, ) = ( −α) α> Let X kn n≥1 for 0, and consider # ( ) 1 ( ) 1 µ = µ (ε ) = meas x n − ε , x n + ε ∩ [0, 1] n n n j 2 n j 2 n 0≤ j≤kn where meas is the Lebesgue measure on R. The degree of resolution de X via ε is deﬁned by

~~log(µ /ε ) g(ε, X) = lim sup n n , n→∞ log kn while the resolution function on R+ is deﬁned by~~

~~1 ρ(α) = ρ(α, X) = π(ε(α), X) = g(ε(α), X), ( ) α 262 where log(µ /ε ) π(ε, X) = lim sup n n n→∞ log(1/εn)~~

~~Since µn ≤ min{1,εnkn}, one has 1 g(ε, X) ≤ { , s(ε)}, ρ(α, X) ≤ , (α > )() min 1 min 1 α 0 263~~

~~When the function ρ(α) is constant on an interval (0,α0], then the value of ρ(α) for 0 <α≤ α0, will be called as the fractal dimension of X.By(263), clearly,~~

~~0 ≤ dim fX≤ 1~~

~~367 CHAPTER 4~~

~~On the other hand, in order to introduce another dimension, for α ∈ [0, 1] put (α, ) = ( (n) − (n))α Hn Xn x j+1 x j 0≤ j~~

Aconnection between the Hausdorff dimension and the resolution degree is given by: g(ε, X) ≤ s(ε, X) dim X (265) for all X and ε. Moreover, sup ρ(α, X) = dim X α>0 Let now log d D = : d|n (n = 2, 3,...) (266) n log n

~~A subsequence {Dn : n ∈ A} of (266) will give a system~~

~~D = D(A)~~

Then Mendes` France and Tenenbaum prove that there exists a sequence A of density 1 such that for all α>0, 1 ρ(α, D(A)) = , ( ) min log 2 α 267 so there exists a sequence A of density 1 such that

~~dim D(A) = log 2 (268)~~

~~Particularly, from (267), (268) it results that for a suitable sequence A of density 1~~

~~dim fD(A) = dim D(A) = log 2 (269)~~

If the distribution of the prime divisors of a typical integer n were sufﬁciently regular, then this dimension would be 1; the fact that this dimension is strictly less than 1 is therefore evidence of some degree of irregularity in the distribution of the prime divisors.

~~368 SPECIAL ARITHMETIC FUNCTIONS~~

15 The divisor graphs In what follows we will study graphs connected to the set of divisors of a number. Given a number n one can generate a graph D(n) that reﬂects the structure of divisors of n as follows. The vertices of the graph represent all the divisors of n,each vertex is labelled by a certain divisor. If r and s are two divisors of n and r > s, then there is an edge between the vertices s and r iff s divides r and the ratio r/s is a prime number (see K. R. Bhutani and A. B. Levin [26]). As in the theory of graceful graphs (see J. A. Gallian [140] for a survey on various graph labelings), we label such an edge by the difference r − s of the labels of its vertices. Let SD(n) and SD(n) denote the sum of the labels of all edges, and all edges except the ones terminating at n, respectively. It is immediate that n SD(n) = SD(n) − n − (270) p|n p

~~= a1 ... ar where p denotes a prime. If n p1 pr is the prime factorization of n, then~~

~~a +1 r p j − 1 SD(n) = (pai − 1) j (271) i − i=1 1≤ j≤k, j =i p j 1~~

~~A number n is called graceful if~~

~~SD(n) = n (272)~~

In [26] it is shown that n is graceful iff n = 4q,where q is an odd prime. (273) As a corollary, the only perfect number (see Chapter 1), which is graceful is 28. In a series of papers, beginning from 1995, E. Saias has studied another graph, called as the divisorial graph the graph of a relation R f deﬁned on the positive integers by

~~aRf b ⇔ a|b or b|a (274) x ≤ Let R f be the restriction of R f to the integers x.For such integers, another x relation Rg will be deﬁned as~~

x ⇔ , ≤ ( ) aRg b [a b] x 275 where [a, b] denotes the l.c.m. of a and b. x = , ,..., − = = ,..., If ni R f ni+1 for i 1 2 k 1 and ni n j for i j, then n1 nk is x ( ) said to be a chain of length k for the relation R f . Let f x denote the maximum

~~369 CHAPTER 4~~

x value of k, and deﬁne similarly a quantity for the relation Rg .In[293] Saies proved that cx/ log x ≤ f (x) ≤ g(x) ≤ cx/ log x, x ≥ 2,(276) for certain positive constant c and c. In [113] Erdos¨ and Saias considered the minimum number φ f (x) of chains for x ≤ the relation R f that are required in order that every positive integer x belongs to φ ( ) x at least one such chain, and the corresponding number g x for the relation Rg . The x , x analogues quantity when the chains for R f Rg are pairwise distinct is denoted by φ∗ ( ), φ∗( ) f x g x , respectively. The following results hold true: c x c x 1 ≤ φ (x) ≤ φ (x) ≤ 2 , x ≥ 2 (277) log x g f log x x c x ≤ φ∗(x) ≤ φ∗ (x) ≤ , x ≥ 2 (278) 3 g f 2 Let R(x) denote the maximum number of integers r ≥ 2 which can be written as n + n = i 1 = i x r or r for R f . Then in [294] the double-inequality ni ni+1 √ √ (1 − o(1)) 8x ≤ R(x)< 8x (279) is proved. Let f (x, y) (resp. g(x, y))denote the maximum number of a union of y chains x x of R f (resp. Rg ). In a recent paper, Saias [295] shows that log 2y log 2y c x ≤ f (x, y) ≤ g(x, y) ≤ c x for x ≥ y ≥ 1,(280) 3 log 2x 4 log 2x with c3, c4 > 0 constants. x ≤ For the graph Rg ,at-chain of integers x of length k is a k-tuplet (a1, a2,...,ak) of integers ≤ x, ai = a j for i = j, such that for all 1 ≤ i < k one has [ai , ai+1] ≤ xt. Let h(x, t) be the maximum length of a t-chain of integers ≤ x. Then log 2t log 2t c x ≤ h(x, t) ≤ c x for x ≥ t ≥ 1,(281) 5 log 2x 6 log 2x where c5, c6 are positive constants. This improves a result of C. Pomerance [277] to the effect that h(x, xa) lim lim sup = 0 (282) + a→0 x→∞ x

~~370 SPECIAL ARITHMETIC FUNCTIONS~~

~~We note that Erdos,¨ Freud and Hegyvari´ [105] have shown that for all function o(1) one has h(x, x1−o(1)) ∼ x as x →∞ (283)~~

~~4.3 Arithmetic functions associated to the digits of a number~~

~~1 The average order of the sum-of-digits function~~

Let q > 1beafixed integer and denote by sq (n) the sum of digits of n written in base q, i.e. m sq (n) = ai (1) i=0 m i where n = ai q for some integer m and 0 ≤ ai < q, am > 0. We will write i=0 s(n) = s10(n) for the decimal expansion (i.e. when q = 10). This function occurs in many problems of Number theory or Mathematics. It is sufficient to mention the well known Legendre formula giving the exponent ep(n!) of a prime p in n!(see e.g. [177]): n − s (n) e (n!) = p (2) p p − 1 Forasimilar result, which can be attributed to Kummer (see J. W. Sander [298]) we note that 2n e = s (n), (3) 2 n 2 2n where denotes a binomial coefficient. (For prime factors of binomial coeffi- n cients or consecutive integers, see [260], Chapter 12). It was L. E. Bush [46] who first showed that q − 1 sq (n) ∼ x log x as x →∞ (4) n≤x 2 log q In 1949 L. Mirsky [259] proved that q − 1 sq (n) = x log x + O(x), (5) n≤x 2 log q which is best possible as can be seen by taking x = (1 − q)ql .Aweaker error term, namely O(x log log x) was discovered a year earlier by R. Bellman and H. N.

~~371 CHAPTER 4~~

Shapiro [20]. P. Cheo and S. Yien [54] have rediscovered (5), and further showed that if bm(x) = card{y ≤ x : sq (y) = m}, then 1 log x m b (x) ∼ as x →∞ (6) m m! log k M. P. Drazin and J. S. Grifﬁth [76] have studied more generally the sum of kth powers of digits of n written in base q: m ( ) = k ( ) sq,k n ai 7 i=0 for a positive integer k, and by putting n−1 n−1 1 k Aq,k(n) = sq,k(m), δq,k = m , m=1 q m=1

~~n log n Fq,k(n) − Aq,k(n) Fq,k(n) = δq,k ,q,k(n) = log q nδq,k they proved that for all q, k, n one has:~~

q,k(n) ≥ 0,2,k(n)<1,(8) and for all q > 3, q − 1 log(q − 1) , (n)< · (9) q k q − 2 log q For q = 2(when one obtains the most precise results), these have been rediscovered by M. D. Mc Ilroy [186] as well as by I. Shiokawa [308]. For results on Aq,k(n) when q = 2 and k ≥ 0see also K. B. Stolarsky [319]. In 1968 J. R. Trollope [344] discovered the following result. Let g(x) be periodic of period one and deﬁned on [0, 1] by   1 1  x, if 0 ≤ x ≤  2 2 ( ) = g x   1 1  (1 − x), if < x ≤ 1 2 2 and put ∞ 1 f (x) = g( i x)() i 2 10 i=1 2 Now, if n = 2m(1 + x), where 0 ≤ x < 1, one has − n 1 n log n s2(k) = − E2(n), (11) k=0 2 log 2

372 SPECIAL ARITHMETIC FUNCTIONS where log(1 + x) E (n) = 2m−1 2 f (x) + (1 + x) − 2x . 2 log 2 In 1975 H. Delange [73] has extended by another method Trollope’s results as follows: There exists a function F : R → R of period one, which is continuous and nowhere differentiable, such that

− 1 m1 q − 1 log m sq (n) = log m + F (12) m n=0 2 log q log 2 for all m ≥ 1. In fact, q − 1 F(x) = (1 + [x] − x) + q1+[x]−x h(q x−[x]+1), 2 where ∞ g(qr x) h(x) = r r=0 q and x q − 1 g(x) = [qt] − q[t] − dt 0 2 Delange determined also the explicit Fourier series of F. More generally, for all real numbers x ≥ 1wehave q − 1 log x sq (n) = x log x + xF − h(x) + (1 + [x] − x)sq ([x]), (13) n≤x 2 log q log q where [x] denotes the greatest integer ≤ x. We note that for q = 2, relation (12) implies Trollope’s result (11). C. Cooper and R. E. Kennedy [58] have applied Trollope’s method to extend (11) to the general case: 1 1 Let g(x) be periodic of period and on 0, be equal to the piecewise q −1 q − 1 a (q − a)a linear function connecting the points , where a is a nonnegative q2 − q 2q integer and 0 ≤ a ≤ q. Let

~~∞ 1 f (x) = g(qi x) i i=0 q~~

373 CHAPTER 4 and let n = qm(1 + (q − 1)x), with 0 ≤ x < 1. Then − n 1 q − 1 n log n sq (k) = · − Eq (n), (14) k=0 2 log q where q − 1 E (n) = qm f (x) + (1 + (q − 1)x) log (1 + (q − 1)x)− q 2 q (a − 1)a −a (1 − a + (q − 1)x) − m m , m m 2 m m i m where n = q (1 + (q − 1)x) = ai q = amq + nm−1 is the base q representation i=0 log t of n and log denotes the base q logarithm (in fact log t = ). q q log q In 1986 P. Kirschenhofer [220] showed that the variance 2 1 1 = 2( ) − ( ) Vx s2 n s2 n x n1, and s (n) denotes f (mi ) (where for the integer n, |σ (mi )| i=1 i=1 is the unique admissible representation of n, and f an application from A∗ to R). Dumont and Thomas prove that there exist a real number α and the functions Gk,h with 1 (s f (n) − αl)2k = (2k − 1)(2k − 3)...1 · βklk+ x < n x h + l Gk,h(l) + η(x), (16) h

~~374 SPECIAL ARITHMETIC FUNCTIONS~~

For the odd moments (i.e. 2k − 1inplace of 2k), the ﬁrst term of the right side of (16) disappears. The functions Gk,h are nowhere differentiable if β = 0 and α = f (ω),with ω being the empty word. Result (16) has applications to some models in Theoretical physics. In 1986 J. Coquet [66] established (by using the Delange method) the formula k h 1 k log x log x s2(n) = + Fk,h(l)(17) x n

They conjecture the better error term O(logk−1 x), and in [60] this is established in the special case when x is a power of 10 (i.e. x = 10a). The conjecture is proved to be true by T. C. Brown [43] in the more general case provided x is restricted to the set of those positive integers having at most m nonzero digits in their base 10 representations. $ % q Let L (n) denote the number of ”large” digits (i.e. or more) of n in base q. q 2 Recently Mirsky’s theorem (5), as well as Trollope’s (11) and (14) have been proved

~~375 CHAPTER 4~~

to the function Lq (n) by C. Cooper and R. E. Kennedy [61]: $ % q x log x 2 Lq (n) = + O(x)(21) n≤x q log q $ % q n−1 n log n ( ) = 2 − ∗( ), ( ) Lq k Eq n 22 k=0 q log q ∗( ) where Eq n is deﬁned similarly to that in (14).

~~2 Bounds on the sum-of-digits function Relations (8) and (9) give certain lower and upper bounds for expressions related to the sum of digits function. As in (8) and (9), let~~

n−1 Aq (n) = Aq,1(n) = sq (m), m=1 and put 1 log n S(q, n) = 2 A (n) − (q − 1) n , q 2 log q where [·] denotes the integer part. D. M. E. Foster [127] obtained the upper bound S(q, n) < q − 1,(q ≥ 2)(23) n which is best possible. For q = 2, 3, 4, 5, 7 one has the best possible lower bounds (see [128]) ( , ) ( , ) ( , ) −2 < S 2 n ;−2 < S 3 n ;−9 < S 4 n ; 3 n 7 n 23 n (24) ( , ) ( , ) − 7 < S 5 n ;−6 < S 7 n 13 n 19 n

~~Clearly every positive integer n ≡ 0 (mod q) is of the form n = nm, where~~

~~t0 t0+t1 t0+t1+t2 t0+t1+···+tm nm = a0q + a1q + a2q + ···+amq (25) for some m ≥ 0, t0 = 0, positive integers t1,...,tm and non-zero coefﬁcients a0, a1,...,am ∈{1, 2,...,q − 1}.~~

~~376 SPECIAL ARITHMETIC FUNCTIONS~~

~~For convenience of notation, given such an integer n, introduce~~

~~t1 t1+···+ti n−1 = 0, n0 = a0 and ni = a0 + a1q + ···+ai q for 1 ≤ i ≤ m. Then S(q, nm) has a simple form:~~

~~m m t0+···+tr S(q, nm) = ar (ar − 1)q + {2ar − (q − 1)tr }nr−1 (26) r=0 r=0~~

~~It is easily veriﬁed that for s,apositive integer,~~

( , s ) ( , ) S q q nm = S q nm , s q nm nm so that we may assume that n is of the form (25) for some integer m ≥ 0. The inequality S(q, n ) m + 1 m ≤ (q − 1) 1 − (27) m+1 nm q − 1 appears in [127]. Let βq be the unique odd integer satisfying 2 2 3q − 8q − 9q + 1 ≤ βq ≤ 3q − 8q − 9q + 1 + 2, unless the even integer 8q2 − 9q + 1isaperfect square. Put 2 q − 3q − (βq − 1)(βq − 5) hq = 2(3q − βq + 1) In [128] Foster shows that for all odd q ≥ 9,

~~S(q, nm) > −hq (28) nm For q = 3, 5, or 7 there is a more precise result, namely~~

~~S(q, nm) ≥−hq (m), (29) nm where~~

~~6(3m − 1) (q2 − 3q + 4)qm−1 − 1 h (m) = , h (m) = (q = 5, 7) 3 7 · 3m+1 − 5 q 2{(3q − 2)qm−1 − 1}~~

~~377 CHAPTER 4~~

~~Let~~

∗ 1 m−l+1 m−l+2 m−2 1 m−1 m n = n − + (q − 1)(q + q + ···+q ) + (q − β )q + q , m m l 2 2 q m−l t0+···+tr where nm−l = ar q and q is odd. r=0 If m − l remains ﬁxed, then (see [128]) S(q, n∗ ) m →− ( →∞)() ∗ hq m 30 nm Similar results when q is even are proved by Foster in [129]. By considering the change of the sum of digits by multiplication, put

S(x) = card{n ≤ x : s2(kn) ≥ s2(n) for all k ≥ 1}, K. B. Stolarsky [320] showed e.g. that S(x) 1 lim inf ≤ (31) x→∞ x 2 N Let 3(n) = s2(3n) − s2(n) and put K N (x) = card{n ∈{0, 1,...,2 − 1} such 1/2 −1/2 that 3 3(n)N < x}. Then I. Katai´ [206] proved that for all x ∈ R x −N 1 2 lim 2 K N (x) = exp(−u /2)du (32) N→∞ 2π −∞

~~More generally (32) can be extended to k(n) = s2(kn) − s2(n),where k is not 1/2 apower of two, and with k(n)~~

y1 = given positive integer; yn+1 = yn + s2(yn)(n ≥ 1). Then the following is true: n y = log n + O(n(log n log log n)1/2)(33) n 2 I. Shiokawa [308] proved that 1 x log x lim inf − s2(n) = 0 x→∞ x 2 log 2 n≤x (34) 1 x log x log 3 lim sup − s2(n) = 1 − →∞ x x 2 log 2 n≤x 2 log 2

378 SPECIAL ARITHMETIC FUNCTIONS while A. Stein and I. E. Stux [316] deﬁned A2(x) = s2(s2(n)), n≤x 2 log 2 c(x) = A (x) , 2 x log log x proving that 1 log log log x 3 log log log x + O ≤ c(x) ≤ + O (35) 2 log log x 2 log log x The exponential sum of digit counting function was deﬁned in 1899 by J. W. L. Glaisher [145] as s2(m) φz(x) = z for z = 2. m

Glaisher showed that φ2(n) represents the number of odd integers in the ﬁrst n rows of Pascal’s triangle. (36) log(1 + z) φ (x) Let t = t = and deﬁne ψ (x) = z , where z > 0, z = 1isareal z log 2 z xt number. For z = 2, upper and lower bounds for ψz were obtained by Stolarsky [319] and H. Harborth [175]. In the general case, A. H. Stein [314] proved that

~~0 < lim inf ψz(x) ≤ 1 ≤ lim sup ψz(x)<2 x→∞ x→∞ (37) lim inf ψ (x)~~

~~n n+1 For z = 2ifweletqn = min{ψ2(x) :2 < x < 2 }, then 1 log 3 lim inf ψ2(x) ≥ qn − · (38) x→∞ 2n log 2~~

For properties of the more general exponential sum with s2(m) replaced to sq (m), see A. H. Stein [315]. We note that the case z < 0, particularly z =−1 (involving the Thue-Morse s (n) sequence deﬁned by tn = (−1) 2 (n ≥ 1)) will be studied later.

~~3 The sum of digits of primes In what follows we will study the sum of digits of primes.In1961 W. Sierpinski [310] proved that for each q > 1,~~

~~lim sup sq (pn) =+∞,(39) n→∞~~

~~379 CHAPTER 4~~

~~where pn denotes the nth prime number. (39) implies that~~

~~s1(pn+1)>sq (pn) for inﬁnitely many n (40) In 1962 P. Erdos¨ [104] solves an open question by Sierpinski, namely~~

~~sq (pn)>sq (pn+1) for inﬁnitely many n (41) By assuming Schinzel’s H-Hypothesis, in [310] it is shown also that~~

sq (pn)>sq (pn+1)>sq (pn+2) for inﬁnitely many n; (42) and sq (pn) = sq (pn+1) = sq (pn+2) for inﬁnitely many n (43) A. Schinzel has shown (see [310]) that accepting the H-hypothesis, for each given positive integer m, there exist positive integers n1, n2 and n3 such that ( )< ( )<···< ( ) s1 pn1+1 s1 pn1+2 sq pn1+m ( ) = ( ) = ···= ( ) ( ) sq pn2+1 sq pn2+2 sq pn2+m 44 ( )> ( )>···> ( ) sq pn3+1 sq pn3+2 sq pn3+m It is interesting to note that the proof of (41) is based on a strong theorem on the distribution of primes (Hoheisel-Ingham theorem on π(x),see [260]), and an elementary proof is missing. I. Shiokawa [309] proved that for all q > 1, 1 x 1/2 sq (p) = (q − 1) + O(x(log log x/(log x)) )(45) p≤x 2 log q where p runs through the primes. I. Katai´ [207] was able to show that − k q 1 k −1 sq (p) − log x x(log x) 2 ,(46) p≤x 2 log q where k is a positive integer. For k = 1 this improves relation (45). E. Heppner [180] considered, more generally the values of sq (n) for n ∈ B, where B ⊂ N is a subset of positive integers. Let B(x) be the counting function of B, and assume that log(x/B(x)) = o(log x). Then      1/2   + x   log log x log   q − 1 log x  B(x)   sq (n) = · B(x) 1 + O    2 log q   log x  n≤x,n∈B   (47)

~~380 SPECIAL ARITHMETIC FUNCTIONS~~

For B ={p1, p2,...,pn,...} (i.e. the sequence of primes), all conditions are satisﬁed and (47) gives (45). For the prime values taken by sq (n),R.Warlimont [347] proved that x card{n ≤ x : s (n) = prime} . q log log x

4Niven numbers An integer n is called a Niven number if s(n)|n,(48) i.e. if it is divisible by its (base 10) digital sum. These numbers were first considered by I. Niven [266] at a Conference on number theory. In fact this concept was introduced and investigated by R. Kennedy, T. Goodman and C. Best [217] and R. Kennedy [214]. Some examples of Niven numbers are 8, 12, 180, 4050. The set N of these numbers is clearly infinite since 10k ∈ N for any k = 1, 2,... Let N(x) be the number of elements of N not exceeding x. The natural density of Niven numbers is zero; that is N(x) lim = 0 (49) x→∞ x This was shown in 1984 by R. E. Kennedy and C. N. Cooper [215]. Another proof by them can be found in [62], where general conditions are given for a set to have natural density equal to zero. Let Nk ={n ∈ N : s(n) = k}. Then (see C. N. Cooper and R. E. Kennedy [63]) k Nk(x) ∼ C(k)(log x) (x →∞)(50) which was a partial answer to the determination of the asymptotic formula for N(x). For similar particular results, see [64]. The first nontrivial bounds for N(x),byelementary methods are due to J.-M. De Koninck and N. Doyon [226]: For given ε>0, x log log x x1−ε N(x) (51) log x Later, using complex variables as well as probabilistic arguments, De Koninck, Doyon and Katai´ [228] have deduced that x N(x) = (c + o(1)) (52) log x 14 where c = log 10 = 1.1939 ... 27

~~381 CHAPTER 4~~

~~Let T (x) denote the number of Niven number n ≤ x such that n + 1isalso a Niven number. Recently De Koninck and Doyon [227] showed that~~

~~x log log x T (x) (53) (log x)2~~

~~Let T ={n ∈ N : n + 1 ∈ N}.Then (52) implies~~

~~1 < +∞,(54) n∈T n while (52) implies that~~

1 = (c + o(1)) log log x,(55) n≤x,n∈N n where c is the same as in (52). Consecutive Niven numbers have been first studied by Kennedy [214] where it was shown that no more than 21 consecutive Niven numbers is possible. Later, Cooper and Kennedy [65] replaced 21 with 20, and showed in fact the existence of infinitely many such sequences. This result is best possible. (56) Let q > 1. In 1994 H. G. Grundman [160] defined a q-Niven number n as satisfying

sq (n)|n (57) For q = 10 one obtains the usual Niven numbers. Grundman proved that the length of a sequence of consecutive q-Niven numbers is at most 2q.(58) He found also the smallest sequence of 20 consecutive Niven numbers, and conjectured that there exists a sequence of consecutive q-Niven numbers of length 2q for each q > 1. (59) For q = 2, 3, T. Cai [47] showed that there exists an inﬁnite family of sequences of consecutive q-Niven numbers of length 2n.(60) De Koninck and Doyon [227] denote by nk the smallest positive integer n such that the interval [n, n + k − 1] does not contain any Niven number (k ≥ 1given), and prove: k+3 nk <(100(k + 2)) (61)

~~For each l ∈ [2, 20] let Tl (x) = card{n ≤ x : n + 1, n + 2,...,n + l − 1areall Niven numbers}.DeKoninck and Doyon conjecture that~~

~~( ) x ( ) Tl x l 62 logx~~

~~382 SPECIAL ARITHMETIC FUNCTIONS~~

A number n is called all-Niven (see E. Weisstein [351]) if sq (n)|n for all q > 1. Then A. Kertesz proved that the only all-Niven numbers are 1, 2, 4 and 6. (63) Another special Niven numbers are the so-called sum-product numbers n for which n = s(n) · a(n), where a(n) = product of digits of n. E.G. 135 = (1 + 3 + 5) · (1 · 3 · 5).D.Wilson (see [312] (sequence A038369), [352]) proved that there are only three such numbers, namely 1, 135 and 144. (64)

~~5 Smith numbers~~

~~< = a1 ... ar ( ≥ ) Let 1 n p1 pr r 2 be the prime factorization of the composite number n.A.Wilansky [354] deﬁned n to be a Smith number if~~

s(n) = a1s(p1) + ···+ar s(pr ) = Sp(n)(65) i.e. numbers whose digit sum is equal to the sum of the digits of its prime factors (counted with multiplicity). For example, 493775 = 3 · 52 · 65837 is a Smith number, since 4 + 9 + 3 + 7 + 7 + 5 = 3 + 2 · 5 + 6 + 5 + 8 + 3 + 7(= 42). The first few such numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, ... (see [312], sequence A006753). Wilansky found 360 Smith numbers n ≤ 105, and S. Oltikar and K. Wayland [268] have noted that relatively large Smith numbers are easily generated from primes whose digits are all 0’s or 1’s, but only a small number of such primes are known. In 1987 W. L. Mc Daniel [70] proved the existence of infinitely many Smith numbers. (66) Mc Daniel introduced also the concept of a k-Smith number which satisfies

~~Sp(n) = k · s(n), (67)~~

3 where Sp(n) is defined in (65). For example, n = 104 = 2 · 13 is a 2-Smith number as Sp(104) = 10 = 2 · s(104). The number n = 402 = 2 · 3 · 67 is 3-Smith, and there are 21 2-Smith numbers, three 3-Smith numbers, one 7-Smith number, etc. for n ≤ 1000. There are infinitely many k-Smith numbers (see [70]). (68) For tables on k-Smith numbers, see [312] (e.g. sequences A050224 for k = 2 and A050225 for k = 3). Let Rn be the nth repunit number. It seems that the largest Smith number is (see [353]) 4594 2297 1476 3913210 9 · R1031 · (10 + 3 · 10 + 1) · 10 (69) Forspecial sets of Smith numbers, see S. Yates [356], [357], Mc Daniel [71], Mc Daniel and Yates [72] (where one can find extensions to base q > 1, too).

~~383 CHAPTER 4~~

~~R. L. Bishop [29] says that n is a Smith-Jones number,if~~

~~s(n) ≡ Sp(n)(mod 9)(70)~~

Let p1 and q1 be such numbers that s(p1) = s(q1), and suppose that n = a1 ... ar p1 pr is a Smith-Jones number. = a2 ... ar Then m q1 p2 pr is also a Smith-Jones number. (71) If a composite number n satisﬁes

~~s(n) = s(p1) + ···+s(pr )(72)~~

(in place of (65)), then it is called a hoax number,see [312], sequence A019506. The first few hoax numbers are 22, 58, 84, 85, 94, 136, 160, ... Are there infinitely many? Consecutive Smith numbers (n, n + 1) are also called Smith brothers. The first few Smith brothers are (728, 729), (2964, 2965), (3864, 3865), (4959, 4960), ... It is not know if there exist infinitely many such pairs. By making investigations with a computer, J. L. James [193] conjectures that there do not exist Smith numbers which are perfect numbers. Monica sets and Suzanne sets are defined in M. Smith [313]. For example, let

~~Sn ={m composite : n|s(m) and n|Sp(m)},(73) where Sp is deﬁned by (65). Then Sn is called the nth Suzanne set. Every Suzanne set has an inﬁnite number of elements. (74) Similarly, let~~

~~Mn ={m composite : n|s(m) − Sp(m)}~~

~~Then Mn has an inﬁnite number of elements, Mn ⊂ Sn, and if m is a Smith number, then m ∈ Mn for all n.Ifm is a k-Smith number, then m ∈ Mk−1.(74)~~

~~6 Self numbers In 1963 D. R. Kaprekar [202] introduced the concept of a self-number.Letq > 1beﬁxed integer. Then a positive integer n is called to be a q-self-number if the equation~~

n = m + sq (m)(75) has no solution in positive integers m. Otherwise, it is called a q-generated number. In 1973 V. S. Joshi [200] proved that if q is odd, then n is a q-self number iff n is odd. Every even number in an odd base is a generated number. (76)

~~384 SPECIAL ARITHMETIC FUNCTIONS~~

This property appears also in Solution of Problem E2408, see [282]. The set of q-self numbers is inﬁnite, and its asymptotic density exists, and is positive (when q 1 is odd, by (76) this density is ). (77) 2 More general results are treated by R. Guaraldo [164]. In part II of [164], the above density is explicitly calculated for all bases q > 1. For example, when q = 10, the density is approximately 0.9022222 ... (78) When q = 2, a more precise result has been obtained by U. Zannier [359]:

~~card{n ≤ x, n = 2 − self-number}=Lx + O((log x)s2([x])) (79) where L > 0;~~

~~card{n ≤ x, n = 2 − generated}=Lx + O(log2 x)(80)~~

More general results for the number of solutions of m + sq (m) ≤ x are proved by M. B. S. Laporta and E. Laserra [240]. For example, when b is odd, a is arbitrary, L card{m + s (m) ≤ x, m + s (m) ≡ a (mod b)}= x + O(xµ), (81) q q b where L is as in (80), and 0 <µ<1. R. B. Patel [271] proved that n = 4q + 2isaq-self-number iff 2|q, q ≥ 4; 2q is a q-self-number in every even base q ≥ 4; q2 + 2q + 1isaq-self-number in an even base q ≥ 4; 3q + 1isaq-self-number iff 2|k, k ≥ 4; q2 + 3q + 2isaq-self-number iff 2|q, q ≥ 4. (82) For other classes, see also T. Cai [48]. An integer n is said to be universal generated if it is q-generated for every base q ≥ 2. E.g. 2, 10, 14, 22, 38, 60, ... are such numbers. Let G(x) be the counting function of universal generated numbers. Then (see T. Cai [48]) √ G(x) ≤ 2 x (83)

m i Ageneralization of q-self-numbers can be found in [164], part I. Let n = ai q i=0 be the unique representation of n in base q. Let f (a, i) be a nonnegative, integer- valued function of the digit a, and the place where the digit occurs. m Put s f (n) = f (ai , i), and let i=0

~~T f (n) = n + s f (n), R f ={n : n = T f (x) for some x},~~

~~C f ={n : n = T f (x) for any x}.~~

~~385 CHAPTER 4~~

~~A number n is f -self-number iff n ∈ C f , and it is a f -generated number,iff n ∈ R f . Let f (a, i) satisfy the following conditions:~~

~~f (0, i) = 0 (i = 0, 1, 2 ...) (84)~~

~~f (a, i) = o(qi ) for 1 ≤ a ≤ q − 1 (85)~~

Then the density of f -generated numbers exists. When f depends only on a (1 ≤ a ≤ q − 1), and f (0) = 0, then the density L f exists, and L f < 1ifand only if the application T f is not one-one. (86) It is interesting to note that if one assumes only the second condition of (84), then the density may not exist. Let e.g. f (a, i) = 0ifi is even; = qi if i is odd. Then the density of R f doesn’t exist. Now, a result of different nature is the following ([164], I): If f (a, i) = O(qi /i 2 log2 i) for all a,(87) then the density of R f exists. (88) Similar results hold true, when considering ”factorial base” in place of the q-adic expansion. It is known that (see e.g. G. Faber [120]) n may be uniquely represented in the form n = a1 · 1! + a2 · 2! + ··· + am · m! (0 ≤ ai ≤ i).Let f (a, i) be a nonnegative integer-valued function of i for each ”digit” a,0 ≤ a ≤ i (i = 1, 2,...)and deﬁne s f (n), T f (n), R f , C f as above. Now, in place of (84) assume that (see [164], III) f (0, i) = 0 (i = 1, 2,...)

~~f (a, i) = o(i!) uniformly in i(i.e. sup{ f (a, i), 0 ≤ a ≤ i}=o(i!)). (89)~~

Then the density of R f exists. (90) As a corollary, one obtains that if f (a, i) = F(a) depends only on a, and F(0) = 0, F(a) = o(i!) uniformly in i for all ”digits” a, then the density of R f exists, and this density is < 1iffT f is not one-one. (91) In the case F(a) = a (when T f (n) = n+ sum of ”digits” of n), the density of R f is 0.879888 ... (92) Similarly, assume that

~~f (a, i) = O(i!/i 2 log2 i) uniformly in i (93)~~

~~Then the density of R f exists. (94) More general base expansions will be considered in further sections.~~

~~386 SPECIAL ARITHMETIC FUNCTIONS~~

~~7 The sum-of-digits function in residue classes~~

~~The distribution of sq (n) in residue classes wasﬁrst studied by A. O. Gelfond [141], who for N ∈ N and r ∈ Z considered the sets~~

Sr,m(N) ={n < N : sq (n) ≡ r (mod m)} where q, m ∈ N are ﬁxed positive integers, q ≥ 2, (m, q − 1) = 1. Then 1 1 lim cardSr,m = (95) N→∞ N m for all r, i.e. Sr,m is equidistributed in residue classes mod r.See also M. N. J. Fine [125]. Gelfond conjectured that the joint distribution of sum of digit functions with different bases satisﬁes

{ < ( ) ≡ ( ), ( ) ≡ ( )}= N + ( λ) card n N : sq1 n r1 mod m1 sq2 n r2 mod m2 O N m1m2 (96) for a certain λ<1, where (m1, q1 − 1) = 1, (m2, q2 − 1) = 1 and r1, r2 ∈ Z. In 1972 J. Besineau´ [25] showed that the main term in (96) is true (i.e. the left N side ∼ as N →∞). m1m2 In 1999 D.-H. Kim [218] proved completely Gelfond’s conjecture in the general context of q-additive functions. A function f is called q-additive if f (0) = 0 and f (aqk + b) = f (a) + f (b) for all integers a ≥ 1, k ≥ 1and0≤ b < qk. This notion is due to Gelfond [141]. Let q = (q1,...,ql ) and m = (m1,...,ml ) be tuples of integers with q j , m j ≥ 2 and (qi , q j ) = 1fori = j.Foreach j ∈{1,...,l} let f j be a q j -additive function with integer values and let f = ( f1,... fl ). Further, let F = (F1,...,Fl ) and d = (d1,...,dl ) be deﬁned by Fj = f j (1) and

~~d j = gcd(m j ,(q j − 1)Fj , f j (r) − rFj (2 ≤ r ≤ q j − 1))~~

~~Let f (n) = b (mod m) mean that f j (n) = b j (mod m j ) for each j.Wecalla tuple b of integers admissible with respect to q, m and f if the system of congruences~~

~~Fn ≡ b (mod d) has a solution. Now, Kim’s result can be stated as follows: N/A + O(N λ), if b is admissible, card{n < N : f (n) ≡ b (mod m)}= 0, otherwise (97)~~

~~387 CHAPTER 4~~

~~2 3 2 as N →∞, where λ = 1 − 1/(120l (max q j ) (max m j ) ) and~~

~~A = card{b :0≤ b j < m j , b admissible}.~~

The implied constant depends only on l and q. In 1996 C. Mauduit and A. Sark´ ozy¨ [254] proved an Erdos-Kac¨ theorem for the distribution of sq (n) in residue classes: 1 card{n ∈ Sr,m(N) : ω(n) − log log N ≤ x log log N}−φ(x)= cardSr,m(N) log log log N = O ,(98) log log N where φ(x) denotes the normal law. Other versions can be found in [255] and in J. M. Thuswaldner [336]. Forsystems of q-additive functions, J. M. Thuswaldner and R. F. Tichy [341] deﬁned q, b, m and f as in (97). Let

M(N) ={n < N : f (n) ≡ b (mod m)}, and assume that d = (1,...,1). Then (98) holds for Sr,m(N) replaced with M(N), uniformly for all real x and N ≥ 8. (99) For the distribution in residue classes of the sum of digits function in number ﬁelds, see J. M. Thuswaldner [337]. The function sq (n) in number ﬁelds will be studied later. The theory of q-additive functions of Gelfond has been extended by J. Coquet [67]. Let Q ={Q j } j≥0 be a sequence of strictly increasing positive integers with Q0 = 1 such that Q j |Q j+1 for all j. Note that the sequence Q is uniquely determined by the factors q j = Q j+1/Q j .Isiteasily seen that each nonnegative integer n has a unique ”base-Q”representation of the form n = a j (n)Q j ,where the j≥0 ”digits” a j (n) satisfy 0 ≤ a j (n)

~~388 SPECIAL ARITHMETIC FUNCTIONS~~

Given a system Q, Coquet deﬁned a Q-additive function f : N → C to be a function of the form f (n) = f j (a j (n)) where n = a j (n)Q j is the base- j≥0 j≥0 Q representation of n, and the component functions f j are functions deﬁned on {0, 1,...,q j − 1} and satisfying f j (0) = 0. Set 1 µ j = 1 − max card{0 ≤ n < q j : f j (n) ≡ r (mod m)} and r q j (100) 1 σ j = max card{0 ≤ n < N : f j (n) ≡ 0 (mod m)} 1≤N≤q j N

We call f j uniformly distributed mod m if, for every integer r, 1 1 card{0 ≤ n < q j : f j (n) ≡ r (mod m)}= q j m We say that an arithmetical function f has a limit distribution mod m,iffor each integer r, the limit 1 lim card{0 ≤ n < N : f (n) ≡ r (mod m)} (101) N→∞ N exists. If each of the limits (101) is 1/m,wesay that f has a uniform limit distribution mod m (or that the set in (101) is equidistributed in residue classes mod r (see (95)). The following theorem is due to A. Hoit [183]. Let f be an integer-valued Q- additive function, and let m be a prime. Then the function f has a uniform limit distribution mod m iff at least one of the following conditions holds:

~~(i) For some j, f j is uniformly distributed mod m; ∞ (102) (ii) µ j =+∞ j=0 The function has a (non-uniform) limit distribution mod m iff the following three conditions all hold:~~

~~(i’) There is no j for which f j is uniformly distributed mod m; ∞ µ < ∞; (ii’) j (103) j=0 (iii’) lim σ j = 0, j→∞ where µ j and σ j are deﬁned by (100).~~

~~389 CHAPTER 4~~

Forexample, let f (n) = sum of digits with prime indices in the base-Q representation of n. Then f has a uniform limit distribution mod n.Let f (n) = sum of prime digits in the factorial representation of n. This function has also a uniform 2 limit distribution mod m.IfQ j = (( j + 1)!) , and f (n) = number of digits 1 in this representation, then f has no limit distribution mod n.(In all examples, m = prime.) E. Fouvry and C. Mauduit [130], by studying the problem of existence of almost primes in sets of integers generated by ﬁnite automata, have proved that the sets

{n : s2(n) ≡ r (mod m)} x (m ≥ 1, r integers) contain inﬁnitely many numbers (even below x) with at log x most two prime factors. (104) If M is a set of residue classes (mod m), and cardM ≥ 0.722q, then there are inﬁnitely many primes p such that the residue of s2(p)(mod n) is in M.(105)

8 Thue-Morse and Rudin-Shapiro sequences In 1906 A. Thue [335] (see also J. Berstel [22]) initiated the study of what is now called combinatorics on words with his results on repetitions in words. We say anonempty word w is a square if it can be written in the form xx for some word x. (For example: murmur in English, chercher in French, or mama in Hungarian). We say that w is an overlap if it can be written in the form axaxa for some word x and single symbol a. Thue explicitly constructed an inﬁnite word on two symbols that is overlap-free, that is, contains no subword that is an overlap. He also constructed an inﬁnite word on three symbols that is square-free, i.e. contains no subword that is a square. Thue’s constructions are based on what is now called the Thue-Morse sequence (in terms of its blocks)

t = t(0)t(1)t(2) ···=0110100110010110 ... (106) where t(n) = s2(n)(mod 2) for all n ≥ 0. Thue proved that t is overlap-free. Since Thue’s pioneering work, many other investigators have studied overlap-free words or their generalizations. Thue’s construction was rediscovered by M. Morse [261], who used it in a construction in differential geometry. The chess master M. Euwe [119] rediscovered Thue’s construction in connection with a problem about infinite chess games. In 1980 E. D. Fife [124] described all infinite overlap-free binary sequences. P. Seébold´ [304] proved the remarkable result that t is essentially

~~390 SPECIAL ARITHMETIC FUNCTIONS the only inﬁnite overlap-free binary sequence which is generated by iterating a morphism. (107) J. P. Allouche and J. Shallit [9] proved that the sequence~~

tq,m = (sq (n)(mod m))n≥0 over the alphabet m ={0, 1,...,m −1} is overlap-free if and only if m ≥ q (where q ≥ 2, m ≥ 1 are integers). (108) A palindrome is a word that is equal to its reversal (e.g. radar). Allouche and Shallit prove also that tq,m always contains arbitrarily long squares. It contains arbitrary long palindromes iff m ≤ 2. (109) The sequence tq,m is ultimately periodic, iff m|(q − 1),(110) aresult due to P. Morton and W. J. Mourant [262]. Overlaps, squares and palindromes in sequences have many applications in other fields. For example, in number theory they aid in proving the transcendence of real numbers whose base q expansion or continued fraction expansion have repetitions (see e.g. S. Ferenczi and C. Mauduit [123], M. Queffelec´ [281]). In statistical physics they are useful for studying the spectrum of certain discrete Schrodinger¨ operators (see e.g. A. Bovier and J.-M. Ghez [33], or A. Hof, O. Knill and B. Simon [182]). For Thur-Morse like sequences, see also J. Grytczuk [162]. Closely related to the Thue-Morse sequence is a sequence c defined to be lexicographically least sequence of positive integers satisfying n ∈ c ⇒ 2n ∈ c. Equivalently, c is defined inductively by c0 = 1and ck + 1, if (ck + 1)/2 ∈ c, ck+1 = (111) ck + 2, otherwise This sequence appeared in a problem by C. Kimberling [219]. Define the sequence (dk) by dk = ck − ck−1 for k ≥ 0 (c−1 = 0) Writing the Thue-Morse sequence in terms of its blocks

d d d d t = 011010011 ···=0 0 1 1 0 2 1 3 ..., ( ) one obtains a sequence dk .Then the following surprizing fact is true (see J. P. Alloche et al. [6]): = ≥ ( ) dk dk for all k 0 112 The generating function of the sequence c is 1 c xk = ( − x2e j )/( − xe j )() k − 1 1 113 k≥0 1 x j≥1

~~391 CHAPTER 4 where 2e j + 1, if j is even e1 = 1, e j+1 = 2e j − 1, otherwise (answering a problem by S. Plouffe and P. Zimmermann [275]). In n ∈ c, let r(n) be its rank, i.e. satisfying~~

~~cr(n) = n~~

Then 2n r(n) = + O(log n), (114) 3 2n and r(n) takes the value inﬁnitely often. 3 As a corollary, one has (see [6]): 3k c = + O(log k), (115) k 2 3k and c = inﬁnitely often. k 2 Forarecent new generalization of the sequence t, see U. Astudillo [12]. Let now ( ) = (− )s2 n , tn 1 and for any positive integer k and i ∈ Z denote ( ) = Sk,i n t j 0≤ j0 (116) More precisely, he proved that α 3 S , (n) log 3 < 3 0 < 5 · 3α with α = 20 nα log 4

In 1983 J. Coquet [68] provided an explicit precise formula for S3,0(n) by the use of a continuous, nowhere differentiable function ψ3(x),ofperiod 1: α log n ε(n) S , (n) = n ψ − ,(117) 3 0 3 log 4 3 log 3 where α = ,andε(n) ∈{−1, 0, +1}. log 4

~~392 SPECIAL ARITHMETIC FUNCTIONS~~

The general case has been investigated by J. M. Dumont [91], and S. Goldstein, K. A. Kelly and E. R. Speer [146]. For example, when k = p (prime), in [146] it is proved that by denoting by Sk(n) the column vector with entries Sk,i (n), one has log n S (n) = nαφ + ε(n), (118) k rslog 2

log λ where φ : R → Rp is continuous of period 1, and α = 1 , where λ is the s log 2 1 s largest eigenvalue of a certain matrix M satisfying Sk(2 n) = MSk(n) with s being the multiplicative order of 2 (mod p). Moreover, ε(n) = O(nβ ) with β<α. By using the method of Goldstein, Kelly and Speer, in 1999 M. Drmota and M. Skalba [86] have studied the sum s2( j) Sk,i (y, n) = y j

t Sk(y, n) = (Sk,0(y, n), ...,Sk,k−1(y, n)) denote the vector of Sk,i (y, n). Let e0,...,ek−1 denote the canonical basis of the k- k dimensional vector space C , and let T denote the matrix deﬁned by Tei = ei+1 (ek = e0). The identity matrix is denoted by I . Let M(y) be deﬁned by

s−1 m M(y) = (I + yT2 ) m=0 Then one has s Sq (y, 2 n) = M(y)Sq (y, n)(119) ζ l ( ≤ < ) The eingenvalues of T are exactly the k-th roots of unity k 0 l k with k−1 v = ζ −il ( ) corresponding eigenvectors l k ei ,which are orthogonal. Since M y is a i=0 polynomial in T , the eigenvalues of M(y) are given by

~~s−1 λ ( ) = ( + ζ l2m ) l y 1 y k m=0~~

~~∗ Let l = l2 be the multiplicative subgroup of (Z/kZ) of order ord(k/(k,l))(2). It is clear that λl (y) = λl (y) iff l 2=l2.Wewrite λl (y) instead of λl (y) if~~

393 CHAPTER 4 l ∈ l. Let L denote the system of equivalence classes l = l2.Thenabasis of the eigenspace Vl corresponding to λl (y), l ∈ L,isgiven by vl , l ∈ l. All these eigenspaces are orthogonal. Let Pl (l ∈ L) denote the orthogonal projection on Vl . Furthermore, let V (0) denote the eigenspace corresponding to the eigenvalue 0 (if 0 is an eigenvalue), V (s) the subspace corresponding to eigenvalues of modulus < 1, V (1) the subspace corresponding to those of modulus 1, V (u) corresponding to those with modulus > 1, and V (m) that corresponding to those eigenvalues with maximal modulus. Furthermore, let P(0), P(s), P(1), P(u) and P(m) denote the orthogonal projections on V (0), V (s), V (1), V (u) and V (m),respectively. Then there exists a continuous function F(y, ·) : R+ → V (u) satisfying

~~F(y, 2s x) = M(y)F(y, x)(x > 0), and Pu Sq (y, n) = F(y, n). Consequently O(1) if V (1) ={0}, S (y, n) = F(y, n) + q O(log n) if V (1) ={0}~~

−t st Let |λl (y)| > 1. Then Gl (y, t) = λl (y) Pl F(y, 2 ) is a continuous function log λl (y) G (y, ·) : R → V which satisﬁes G (y, t + 1) = G (y, t).With α (y) = l 1 l l l s log 2 we obtain the fractal representation for Sq (y, n): α ( ) log n S (y, n) = n l y G y, + O(log n)(120) q l s log 2 |λl (y)|>1

~~Let~~

Ak,i,r,m(n) = card{ j < n : j ≡ i (mod k), s2( j) ≡ m (mod r)} and Ar,m(n) = card{ j < n : s2( j) ≡ m (mod r)} Then 1 A , , , (n) = A , (n) + R , , , (n) k i r m k r m k i r m Newman’s theorem (116) can be rewritten as

~~A3,0,2,0(n)>A3,0,2,1(n)~~

~~394 SPECIAL ARITHMETIC FUNCTIONS~~

~~Dumont proved also that (see [91])~~

~~A3,1,2,0(n)~~

Drmota and Skalba [86] discuss the following two kinds of positivity phenomena: (N1) Ak,0,r,0(n)> max Ak,0,r,m(n) for almost all n ≥ 0; 00 for almost all n ≥ 0. They prove the following results: Let k be an odd multiple of 7, and suppose that r = 3. Then (N1) and (N2) hold. (121) Let k, r be positive integers such that k is odd, and r ≥ 2. If s = ordk(2) and r are coprime or if there exists an integer r > 0 such that λ (ζ m)r < λ (ζ m) < < < < l r 0 for those l r ,0 l k,0 m r,with minimal modulus, then (N1) and (N2) fail. (122) Let Pt (t ≥ 1) denote the set of those primes p where the order ord p(2) of2is p − 1 equal to . t For any r > 1 there exists a constant Cr > 0such that for any t ≥ 1primes q ∈ Pt satisfying (N1) or (N2) are bounded by

~~4 4 q ≤ Cr t log t (123)~~

~~The inequality~~

~~A2r −1,0,r,0(n)> max A2r −1,0,r,m(n) for almost all n (124) 0~~

~~R r − , , , (n)<0for inﬁnitely many n ≥ 0, but 2 1 0 r 0 (125) R127,0,7,0(n)>0 for almost all n ≥ 0~~

For the positivity of Sk,0(n),wehave: N Suppose k is divisible by 3, or k = 4 + 1. Then Sk,0(n)>0for almost all n.The only primes p ≤ 1000 satisfying Sp,0(n)>0for almost all n are p = 3, 5, 17, 43, 257, 683. There exists a constant C > 0 such that for any t ≥ 1 and primes p ∈ Pt satisfying Sp,0(n)>0 for almost all n are bounded by

~~p ≤ Ct2 log2 t.~~

~~Furthermore, the total number of primes p ≤ x with Sp,0(n)>0for almost all n is o(x/ log x) as x →∞.(126)~~

~~395 CHAPTER 4~~

( ) n + 1 n + 1 For results on the twisted sum (−1)s2 n , where is a p p Legendre symbol, see another paper by Drmota and Skalba [87]. P. J. Grabner, T. Herendi and R. F. Tichy [150] study the case p = 17, and use these functions to generate a special code: the analysis of this code shows that it is exponential, extremely redundant and thus strongly error correcting with a linear decoding procedure. The Rudin-Schapiro sequence (a(n)) is deﬁned by

~~a(k) = (−1)e(k)~~

s−1 s i where e(k) = εi εi+1 and k = εi e with εi ∈{0, 1} (i.e. the number of con- i=0 i=0 secutive 11-s in the binary expansion of k). In fact, a(2k) = a(k), a(2k + 1) = (−1)ka(k) for k ≥ 0, where a(0) = 1. J. Brillhart and P. Morton [42] have studied the sequences

n n A(n) = a(n), B(n) = (−1)ka(n). k=0 k=0 They proved that " 3 A(n) √ B(n) √ < √ < 6, 0 ≤ √ < 3 (n ≥ 1)(127) 5 n n " A(n) B(n) 3 √ and that the sequences √ , √ are dense in the intervals , 6 , resp. 5 √ n n [0, 3]. (128) J. Brillhart, P. Erdos¨ and P. Morton [41] have deﬁned the continuous versions by

~~[x] [x] A(x) = a(k), B(x) = (−1)ka(k) k=0 k=0 They deﬁned also the functions~~

A(4k x) B(4k x) λ(x) = lim √ ,µ(x) = lim √ k→∞ 4k x k→∞ 4k x which√ are deﬁned for x > 0. These limits exist for all x > 0, and one has µ(x) = 2λ(2x) − λ(x) and µ(4x) = µ(x).(129) The functions λ and µ are continuous on (0, ∞)√,and in√ fact the function λ maps both intervals (0, ∞) and [1, 4] continuously onto [ 3/5, 6].

~~396 SPECIAL ARITHMETIC FUNCTIONS~~

√ The function µ maps (0, ∞) and [1, 4] continuously onto [0, 3]. A real number n x0 > 0iscalled normal (to the base 4) iff the sequence (4 x0)n≥0 is uniformly distributed modulo 1. If x0 > 0isnormal, then λ is not differentiable at x0. Therefore, λ is non- differentiable almost everywhere. (130) The cumulative distribution function is deﬁned as follows. Let (un) be a sequence of real numbers contained in an interval J, and let α ∈ J.IfD(x,α)denotes 1 the number of n ≤ x for which un ≤ α, and if the limit lim D(x,α) = D(α) ex- x→∞ x ists, then the sequence (un) is said to have the distribution D(α) at α. D(α) is called the cumulative distribution function of (un). A(n) For the sequence √ the following is true: n A(n) The cumulative distribution function of √ doesn’t exist at any point of " n 3 √ B(n) √ , 6 . Similar result is true for √ on (0, 3).(131) 5 n The logarithmic distribution function is deﬁned similarly by introducing L(x,α)= 1/n

1≤n≤x,un≤α and considering 1 lim L(x,α)= L(α). x→∞ log x If this limit exists, then it is called the logarithmic distribution of the sequence (un). Now, the" following can be proved (see [41]): 3 √ A(n) If α ∈ , 6 , then the logarithmic distribution of √ exists at α, and 5 n has the value 1 1 L(α) = dx, log u Eα x where Eα ={1 ≤ x ≤ 4: λ(α) ≤ α}. √ B(n) Similar results holds true for α ∈ [0, 3] for √ : n 1 1 L∗(α) = dx, ∗ log 4 Eα x ∗ where Eα ={1 ≤ x ≤ 4: µ(x) ≤ α}.(132)

~~397 CHAPTER 4~~

Finally we mention that the function λ(x) has the logarithmic Fourier series expansion ∞ πinlog x/ log 2 λ(x) = cne (x > 0), n=−∞ where 1 4 λ(x) 1 πni c = ,γ= + , n 1/2+γ n log 4 1 x n 2 log 2 and where the inﬁnite series converges in the (C, 1) sense for all x > 0 ∞ πinlog x/ log 2 1 (i.e. cne = lim (σ0 + σ1 + ···+σk) with k→∞ + n=−∞ k 1 k πinlog x/ log 2 σk = cne ). (133) n=−k In [7], Allouche, Cohem, Mendes` France and Shallit have proved that for all x = 1, |x|≤1 (x ∈ C),

(2n + 1)2 − log 2 xe(n) = . log ( + )( + ) − n≥0 n 1 4n 1 1 x If T (n) = xe(k), then on a nonreal, positive ray of the unit circle, one has 0≤k≤n uniformly T (n) = O(nα) for 0 <α<1. Forexample, if x =−1 one obtains the inﬁnite product result:

~~∞ (− )e(n) (2n + 1)2 1 = −1/2. ( + )( + ) 2 n=0 n 1 4n 1~~

~~See also 15. for related identities. In 1972 E. Zeckendorf [360] proved that any non-negative integer n can be uniquely represented as the sum of distinct Fibonacci numbers~~

z(n) = ( ) n Fk j 134 j=1 where k1 ≥ 2 and k j+1 > k j + 1 ( j ≥ 1). Here F0 = 0, F1 = 1, Fk = Fk−1 + Fk−2 (k ≥ 2). The number z(n) of summands in the Zeckendorf representation will be called Zeckendorf sum-of-digits function; and the sequence (−1)z(n) will be the Zeckendorf

~~398 SPECIAL ARITHMETIC FUNCTIONS~~

Thue-Morse sequence. Let z(n) z(n) S(N) = (−1) and Sk,i (N) = (−1) . n0, x 1 1 2 card{n < N : S(n) ≤ x c log N}=√ e−t /2dt+ N 2π −∞ − 1 +ε log N 2 +O (135) log γ √ √ 1 + 5 5 + 3 5 uniformly for all x ∈ R (as N →∞), where γ = and c = . 2 30 Similarly, 2 N −m − 1 +ε card{n < N : S(n) = m}= exp + O((lg N) 2 ) , 2πc lg n 2c lg N (136) log N where lg N = . log γ Furthermore, 1 1 S(N 1 S(n) = G lg N − + O and N n 3isodd and not divisible by 3. Then α 1 β S , (N) = N k G , lg N + O(N k )(138) k i k i 4k as N →∞, where 0 <αk < 1, βk <αk,andGk,i (x) is a real continuous periodic function with period 1.

~~399 CHAPTER 4~~

~~Drmota and Skalba conjecture that~~

αk Gk,i (x + h) − Gk,i (x) = (|h| ) as h → 0,(139) which would imply that Gk,i is nowhere differentiable. When k = 3, a stronger relation than (138) is valid: α 1 1 S , (N) = N G , lg N + S(N) + O(1)(140) 3 i 3 i 8 3 √ log(4 + 15) as N →∞, where α = and G , (x) are real continuous functions 8 log γ 3 i with period 1. Moreover, G3,0(x)>0 for x ∈ [0, 1),and

~~S3,0(N) ≥ 0forall N ≥ 0 (141)~~

This is an analogue of Newman’s theorem (116). For Zeckendorf expansion of polynomial sequences, see M. Drmota and W. Steiner [89]. J. Coquet and P. Van den Bosch [69] considered the unique representation n = a j (n)Fj , where a j (n) ∈{0, 1}, j≥1 and a j (n)a j+1(n) = 0. If SF (n) = a j (n) and S(n) = sF (n), then j≥1 n

~~400 SPECIAL ARITHMETIC FUNCTIONS~~

k where αk = α + k − 1, α being the greatest positive root of the polynomial√ P(x) = 1 5 − 5 xk − xk−1 − 1. For k = 1, α = ; while for k = 2 one has α = ,aresult 1 2 2 10 due to P. H. St. John [196]. J. M. Dumont [92], by applying the theory of substitutive numeration systems, has obtained another proof of (142). He showed also that 2 lim (wk(u) − nα2) /(ncardFk(n)) = βk (143) n→∞ u∈Fk (n) √ 5 − 5 α2 · αk where α = and β = k . 2 10 k k(α − 1) + 1

9 q-additive and q-multiplicative functions We now study more closely the q-additive and q-multiplicative functions.(For some results, see also 7.) Recall that a real-valued function f , deﬁned on the non-negative integers, is said to be q-additive if f (0) = 0 and j j f (n) = f (aq, j (n)q ) for n = aq, j (n)q , j≥0 j≥0 where aq, j (n) ∈ Eq ={0, 1,...,q − 1}.Aspecial q-additive function is sq (n). The most general result concerning the mean value of q-additive functions is due to E. Manstaviciusˇ [252]: Let 1 k 2 1 2 k m , = f (cq ), m = f (cq ) k q q 2,k,q q c∈Eq c∈Eq and [logq x] [logq x] ( ) = , 2( ) = 2 Mq x mk,q Bq x m2,k,q k=0 k=0 Then 1 ( ( ) − ( ))2 ≤ 2( ), ( ) f n Mq x cBq x 144 x n≤x which implies 1 f (n) = Mq (x) + O(Bq (x)) x n≤x There also exist distributional results for q-additive functions. In 1972 H. De- lange [74] proved an analogue of the Erdos-Wintner¨ theorem:

~~401 CHAPTER 4~~

~~There exists a distribution function F(y) such that, as x →∞ 1 card{n < x : f (n)~~

~~j max | f (cq )|=o(Bq (x)) cq j~~

[logq x] 2( ) = σ 2 Dq x k,q k=0 and 1 σ 2 = f 2(cqk) − m2 . k,q q k,q c∈Eq Then, as x →∞, 1 f (n) − M (x) card n < x : q < y → φ(y), (147) x Dq (x) where φ is the normal distribution function. Furthermore, N. L. Bassily and I. Katai´ [19] studied the distribution of q-additive functions on polynomial sequences: j Let f be a q-additive function such that f (cq ) = O(1) as j →∞and c ∈ Eq . D (x) Assume that q →∞as x →∞for some η>0 and let P(x) be a polynomial (log x)η with integer coefﬁcients, of degree r and positive leading term. Then, as x →∞ 1 f (P(n)) − M (xr ) card n < x q < y → φ(y)() : r 148 x Dq (x )

402 SPECIAL ARITHMETIC FUNCTIONS and 1 f (P(p)) − M (xr ) card p < x q < y → φ(y)() : r 149 π(x) Dq (x ) (here p denotes a prime). Let q1, q2,...,qd > 1be pairwisely coprime integers, and consider the ql - additive functions fl (n)(l = 1, 2,...,d).Forthe joint distribution of these functions A. J. Hildebrand (see [78]) proved that

1 card{n < x : f (n) →∞ if Bql and Bql x Bql x for every 0, as x .(Note that the function sq (n) is not covered by this result.) M. Drmota [78] has extended the Bassily-Katai´ theorem to the joint distribution case: Let q1,...,qd > 1bepairwisely coprime integers and let fl (1 ≤ l ≤ d) be ( j ) = ( ) →∞ ∈ ql -additive functions such that fl cql O 1 as j and c El .Assume that ( )/( )η →∞ →∞( ≤ ≤ ) η> ( ) Dql x log x as x 1 l d , for some 0, and let Pl x be polynomials with integer coefﬁcients of different degrees rl and positive leading term (1 ≤ l ≤ d). Then as x →∞, ( ( )) − ( rl ) 1 fl Pl n Mql x card n < x : < yl , 1 ≤ l ≤ d → φ(y1)...φ(yd ), ( rl ) x Dql x (152) and ( ( )) − ( rl ) 1 fl Pl p Mql x card p < x : < yl , 1 ≤ l ≤ d → φ(y1)...φ(yd ) π( ) ( rl ) x Dql x (153) This theorem contains the condition that all polynomials Pl (x) have different degrees. It seems that this condition is not necessary. For the case d = 2 and linear polynomials Pl (x) = Al x + Bl , the following result holds true (see [78]): , > ( = , ) ( j ) = ( ) Let q1 q2 1becoprime, fl be ql -additive l 1 2 ,with fl cql O 1 →∞ ∈ ( )/( )η →∞ →∞ as j and c El .Assume further that Dql x log x as x (l = 1, 2) for some η>0. Let Pl (x) = Al x + Bl (l = 1, 2) be arbitrary linear

~~403 CHAPTER 4~~

polynomials with integer coefﬁcients and positive leading terms (Al , ql ) = 1. Then as x →∞, 1 f (P (n)) − M (x) card n < x l l ql < y , l = , → φ(y )φ(y )() : ( ) l 1 2 1 2 154 x Dql x

~~For the sum-of-digits function sq (n),alocal version of (154) holds true: Let (q1, q2) = 1 and set d = gcd(q1 − 1, q2 − 1).Then, as x →∞,~~

1 card{n < x : s (n) = k , s (n) = k }= x q1 1 q2 2    − 2 − q 1 2   kl logq x   1  2 l  1 = d " exp −  + o  2 −  q2 − 1  log x l=1 ql 1 · l 2π · log x 2 logq x 12 ql 12 l (155) uniformly for all integers k1, k2 ≥ 0 with k1 ≡ k2 (mod d). ( ) ≡ ( ( − )) ≡ ( ) Note that sql n n mod ql 1 ,sowealwayshavesq1 sq2 mod d , and consequently

~~{ < ( ) = , = }= , ≡ ( ). ( ) card n x : sq1 n k1 sq2 k2 0 if k1 k2 mod d 156~~

For digital expansion with respect to different bases see also M. Drmota and J. Schoissengeier [85]. For the joint distribution of Q-additive functions on polynomials over ﬁnite ﬁelds, see M. Drmota and G. Gutenbrunner [83]. Recently B. Gittenberger and J. M. Thuswaldner [143] have extended the Bassily- Katai´ theorem to canonical number systems in the ring of Gaussian integers Z[i]. Let b ∈ Z[i]andN ={0, 1,...,|b|2 − 1}.Apair (b, N ) is called a canonical number system if any γ ∈ Z[i]has a representation of the form

h γ = c0 + c1b + ···+chb , ch = 0ifh = 0, where h ≥ 0isanonnegative integer and c j ∈ N for j = 0,...,h. Here b is called base and N is called set of digits of (b, N ). Let (b, N ) be a canonical number system in Z[i]. A function f is called b- additive if f (0) = 0and j j f (γ ) = f (a j (γ )b ) for γ = a j (γ )b (a j (γ ) ∈ N ) j≥0 j≥0

~~404 SPECIAL ARITHMETIC FUNCTIONS~~

The Bassily-Katai´ theorem for b-additive functions can be stated as follows: Let f be a b-additive function such that f (cb j ) = O(1) for j ∈ N and c ∈ N . Further- more, let 1 1 m = f (cbk), σ 2 = f 2(cbk) − m2, k |b|2 k |b|2 k c∈N c∈N L L ( ) = , 2( ) = σ 2, = . M N mk D N k with L [log|b| N] k=0 k=0 1/3 r Assume that D(N)/(log N) →∞as N →∞, and let P(z) = pr z + ···+ p1z + p0 be a polynomial with coefficients in Z[i]. Then, as N →∞, 1 f (P(z)) − M(N r ) card |z|2 < N : < y → φ(y)(157) card{z : |z|2 < N} D(N r ) where φ is the normal distribution function and z runs over the Gaussian integers. Let sb(z) = c0 + c1 + ···+ch be the sum-of-digits function in Z[i]. As a corollary of (157), one has: 1 s (P(z)) − M(N r ) card |z|2 < N : b < y → φ(y)(158) card{z : |z|2 < N} D(N r ) The proof is based among others on certain results on exponential sums over number fields, as well as a two-dimensional version of the Erdos-Tur¨ an-Koksma´ inequality of discrepancy theory. We note that the notion of canonical number system in Z[i]isdue to I. Katai´ and J. Szabo´ [212] who showed that the only bases are given by b =−n ± i, where n = 1, 2,... I. Katai´ and B. Kovacs´ [211] extended the notion to imaginary quadratic fields, while in 1992 I. Katai´ and I. Kornyei¨ [210] considered the general case of algebraic number fields. Afunction f is called q-multiplicative if f (0) = 1 and j j f (n) = f (aq, j (n)q ) for n = aq, j (n)q , j≥1 j≥0 where aq, j (n) ∈ Eq . For q-multiplicative function f satisfying | f (n)|=1, K. H. Indlekofer and I. Katai´ [188] obtained the following result: There exists a constant c1 such that if f is q-multiplicative, with | f (n)|=1 (n ≥ 0) and f (p) = constant for every large prime p, then f k(nq) = 1 for every n ≥ 0, with a suitable integer k ∈ [1, c1]. (159)

~~405 CHAPTER 4~~

The result remains true also if we assume the fulﬁlment of f (p) = constant with the possible exception of primes of a set of relative density zero. q-multiplicative functions of modulus at most 1 have been ﬁrst studied (in the context of Q-multiplicative functions, see 7.) by J. Coquet [67]. Let Q = (Q j ) j≥0 be a sequence of strictly increasing positive integers with Q0 = 1such that Q j |Q j+1 for all j. Put q j = Q j+1/Q j . Then every nonnegative integer n has a unique base- Q representation n = a j (n)Q j , where 0 ≤ a j (n)

(i) For some j≥ 0,µj ( f ) = 0, ( ) (ii) The series (1 −|µ j ( f )|) diverges 160 j≥0 The case of Q-multiplicative functions of modulus at most 1, having a non-zero mean value, has been settled recently by A. Hoit [184]: Let f be a Q-multiplicative function satisfying | f (n)|≤1. Then the mean value of f exists and is non-zero if and only if the following three conditions all hold:

~~(a) For each j ≥ 0,µj ( f ) = 0, ( − µ ( f )) , (b) 1 j converges ( ) j≥0 161 (c) lim σ j ( f ) = 0 j→∞~~

~~406 SPECIAL ARITHMETIC FUNCTIONS~~

~~As a corollary of (160) and (161) one can deduce: Let f be as above. Then the value of f exists if and only if at least one of the following three conditions holds:~~

~~(i); (ii); (b) and (c) (162)~~

The mean value is zero if either condition (i) or (ii) holds. Another generalization of q-multiplicative functions is due to J. Feher´ [122]. Let R0, R1,... be a sequence of subsets of nonnegative integers. We say that it is an R-system,ifthe following conditions hold: 1) 0 ∈ Ri and 1 < cardRi < ∞ (i = 0, 1, 2,...), 2) For 0 ≤ i < j, the smallest positive element of Ri is smaller than the smallest positive element of R j , 3) Each n ≥ 0can be uniquely written as

~~s n = r j (r j ∈ R j , s ≥ 0) j=0~~

The system is called monotonic,ifinaddition one has 4) For each 0 ≤ i < j, the largest element of Ri is smaller than the smallest positive element of R j . Similarly, the system is called bounded,ifinaddition to 1)-3) the following holds true: 5) cardR j is bounded ( j ≥ 0). Forexample, let ki (i = 0, 1,...)be a sequence of integers, ki ≥ 2, and furthermore let d0 = 1, di = di−1ki−1 (i ≥ 1), Ni ={0, 1,...,ki − 1}, Ri = di Ni = {0, di ,...,(ki − 1)di } (i ≥ 0). Then Ri (i ≥ 0) is an R-system. This is the so-called ”britannic number system” of N. G. De Bruijn [44]. For ki = q ≥ 2 for all i ≥ 0, one reobtains the q-ary number system. It is easy to see that the monotonic R-systems are exactly the system considered by De Bruijn. A function f deﬁned on nonnegative integers, with complex arguments is called R-multiplicative (with respect to a given R-system), if

~~s f (0) = 1 and f (n) = f (r j ). j=0~~

~~Forexample, f (n) = an (0 = a ∈ C) is R-multiplicative, for all R. k Let P(z) = ak z + ···+a1z + a0 be a polynomial of complex coefﬁcients, and denote by~~

~~P(E) f (n) = ak f (n + k) + ···+a1 f (n + 1) + a0 f (n)~~

~~407 CHAPTER 4~~

Let us consider the following conditions: 1 (1) lim |P(E) f (n)|=0, n→∞ x n≤x (2) |P(E) f (n)|=o(x) as x →∞, n≤x (3) P(E) f (n) = 0 for all n ≥ 0, 1 (4) lim inf | f (n)|=0, x→∞ x n≤x (5) | f (n)|=o(x) as x →∞. n≤x J. Feher´ [122] proves the following theorem: If (1) holds, then either (3) or (4) hold. There exists such an f for which (4) holds, but (5) doesn’t hold. Assuming that the R-system is monotonic and bounded, then (4) and | f (n)|≤1 (n ≥ 0) imply relation (5). (163) For the R-multiplicative solutions of the recurrence P(E) f (n) = 0 the following result is valid: Let R0, R1,... be such an R-system for which R0 ={0, 1,...,d−1} (d ≥ 2). Assume that P is of degree k (1 ≤ k ≤ d), and that P(0) = 0. Then P(E) f (n) = 0iff s ( ) = α ρn ( ) f n j j 164 j=1 s where α j = 1, and ρ j ( j = 1, s) are distinct roots of P with j=1 ρd = ρd = ···=ρd . 1 2 s G. Barat and P. J. Grabner [16] have introduced and studied the block- m j multiplicative functions as follows. Let a j = qq, j (n) be the digits of n = a j q j=0 in the base q expansion. For a positive integer l,anarithmetic function f is called l-block-multiplicative if there exists a function g deﬁned on {0,...,q }l such that 1 m m j f a j q = g(a j , a j+1,...,a j+l−1) j=0 j=0 with a j = 0 for j > m, and g(0, 0,...,0) = 1. Let Bs denote the set of all digital blocks of length s.ForB ∈ Bs, the expression Bn indicates the concatenation of the block B with the digits of n.Foragivenl- block-multiplicative function f , deﬁne the summatory function F(N) = f (n), n

~~408 SPECIAL ARITHMETIC FUNCTIONS and for ﬁxed s introduce the functions~~

~~( ) = ( ( )) , f n f Bn B∈Bs FB(N) = f (Bn), n~~

f (B) f (ABC) = f (AB) f (BC), B ∈ Bs, A, C blocks of arbitrary length; F(BN) = FC (N) + f (CN), C∈Bs C g ∞. Then there exist continuous periodic functions ψ j ( j = 0,...,m − 1) of period s such that

m−1 δ α µ ( ) = i logq N ( ) j ψ ( ) + ( )() f n N e logq N j logq N o N 166 nmax log g ∞, log t , where t = max{|γ | : γ ∈ spec(U) \{λ}}, q s q 1 1 δ = log |λ|, α = arg λ, and m is the dimension of the largest Jordan block s q s associated to λ.(Herespec denotes the set of eigenvalues.)

~~409 CHAPTER 4~~

~~Under the stated assumptions, for a positive-valued function f ,wehave ( ) = δψ( ) + ( µ), ( ) f n N logq N o N 167 n~~

k ( ) ( )... ( ) = δ ( )i ψ ( ) + ( µ)() f n f1 n fk n N logq N i logq N o N 168 n 0, such that (cr , q) = 1, then     2   1  log x   f (P(n)) −  h(b0,...,bl−1) r  x log x n≤x qk 0≤bν ≤q−1 log q 0≤ν≤l−1 and a similar result, when n is replaced by a prime p, when the right side becomes π(x) log x.

~~10 Uniform - and well - distributions of αsq (n)~~

Areal sequence (xn)n≥1 is called uniformly distributed mod 1 (for short u.d. mod 1) if 1 N lim χI ({xn}) = λ(I ) N→∞ N n=1 for all intervals I ⊂ [0, 1], where χI denotes the characteristic function of I , {x}= x − [x]isthe fractional part of x,andλ(I ) is the length of the interval. Equivalently,

~~410 SPECIAL ARITHMETIC FUNCTIONS the discrepancy 1 N DN (xn) = sup χI ({xn}) − λ(I ) ⊂ , I [0 1] N n=1 satisﬁes lim DN (xn) = 0. N→∞~~

The sequence (xn) is called well-distributed mod 1 (for short, w.d. mod 1) if the uniform discrepancy DN (xn) = sup DN (xn+k) k≥0 satisﬁes lim DN (xn) = 0. N→∞ = √ Clearly, every w.d. sequence is u.d. However, the converse is not true, e.g. xn n is u.d. but not w.d. Similar deﬁnitions are valid for higher dimensional sequences. In 1967 M. Mendes` France [257] discovered that the sequence (α · sq (n)) is u.d. mod 1 (α ∈ R \ Q).(169) In 1980 J. Coquet showed that this sequence is in fact w.d. mod 1. More generally, ( ) ( ) if a sequence xn is w.d. mod 1, then the sequence xsq (n) n is w.d. mod 1, too. (170) P. J. Grabner and R. F. Tichy [157] have extended (169) to (α · sG (n)), where sG is the sum of digits function with respect to a linear recurrence G (see the next section for such expansions). For similar result for sQ(n) of base-Q representations, see A. Hoit [184]. R. F. Tichy and G. Turnwald [343] derived estimates for the discrepancy of (αsq (n)),and M. Drmota [79] generalized and sharpened their results. An arithmetical function f is called strongly q-additive if

~~f (a + qb) = f (a) + f (b)(0 ≤ a < q, b ≥ 0)~~

Let α be an irrational number and suppose that for every ε>0 there exists a c(α, ε) constant c(α, ε) > 0such that hα ≥ for all positive integers h (where x hη+ε denotes the nearest distance of x to integers). Let f be a strongly q-additive function which attains only non-negative integers such that there exists 1 ≤ b ≤ q − 1 with f (b)>0. Then for every ε>0,

~~1 D (α f (b)) (171) N (log N)1/(2η)−ε for all N > 1, where the constant implied by depends on q,α,εand f .~~

~~411 CHAPTER 4~~

~~It is also possible to show (see [343] and [241]) that for every irrational α there exists a constant c(q,α, f ) such that~~

~~c(q,α, f ) D (α f (n)) > (N ≥ 2)(172) N (log N)1/2~~

A generalization of (170) is proved also in [79]: Suppose that f (n) is strongly q- additive which attains only non-negative integers such that gcd{0 < j < q : f ( j)> 0}=1, Then, if a sequence (xn) is w.d. mod 1, then (x f (n)) is w.d. mod 1, too. (173) For the higher dimensional analogue of (169), M. Drmota and G. Larcher [84] have proved that if q1,...,qd are pairwisely coprime integers > 1, then the d- dimensional sequence (α ( ), ...,α ( )) 1sq1 n d sqd n is u.d. mod 1 if and only if α1,...,αd are all irrational. (174) The same result for w.d. mod 1 has been proved by P. J. Grabner, P. Liardet and R. F. Tichy (see [155]) via ergodic theory,evenfor Cantor representations. See also Drmota and Tichy [90]. For applications of ergodic theory to digital problems see also e.g. P. Liardet [243], T. Kamae [201], U. Krengel [237], P. J. Grabner and P. Liardet [154]. For the discrepancy of the above d-dimensional sequence, Drmota and Larcher prove: Let α1,...,αd be irrational numbers. Suppose that there exists η>0 and a −η constant c1 > 0 such that for all integers h > 0wehave hα j ≥c1h for all ≤ ≤ (α ( ),...,α ( )) 1 j d. Then the discrepancy DN of the sequence 1sq1 n d sqd n is bounded by / η log log N 1 2 D ≤ c(c ,η,q) (175) N 1 log N with some constant c(c1,η,q) depending on c1,η, and q = (q1,...,qd ). Conversely, if for some η ≥ 1and some constant c2 > 0 there exists j such that −η hα j ≤c2h for inﬁnitely many integers h > 0, then

~~1 D ≥ c(c ,η,q) · (176) N 2 (log N)1/2η~~

for inﬁnitely many N ≥ 0, where c (c2,η,q) depends on c2,ηand q. 4 An analogue of (169) is due to C. Mauduit and J. Rivat [253]: Let 1 ≤ c ≤ be 3 areal number and α an irrational number. Then the sequence

~~c (α · sq ([n ])) (177)~~

412 SPECIAL ARITHMETIC FUNCTIONS is u.d. mod 1. This result follows from a more general theorem on q-multiplicative functions f : 1 f ([nc]) = f (m)mc−1 + O(x1−ε)(178) 1≤n≤x c 1≤m≤xc as x →∞(where ε>0). For q-multiplicative functions with modulus 1, and a kind of uniform distribution, (namely uniform summability), Indlekofer and Katai´ [189] have recently obtained arepresentation theorem. Afunction g is called uniformly summable (a notion due to Indlekofer [187]) if 1 sup | f (n)|→0asK →∞.(179) ≥ x 1 x n≤x,| f (n)|≥K Let g be uniformly summable. We say that α is in the Bohr-Fourier spectrum of g,if 1 lim sup f (n)e(−nα) > 0 (180) →∞ x x n≤x (where e(x) = exp(2πix)). Assume now that f is q-multiplicative with modulus 1, and that g is uniformly summable multiplicative function (in ordinary sense), and that 1 lim sup f (n)g(n) > 0.(181) →∞ x x n≤x Then f (n) can be written as r f (n) = e h(n) D r with a suitable rational number and with a function h, which is q-multiplicative D with modulus 1, for which

∞ q−1 Re(1 − h(cq j )) < ∞ (182) j=0 c=0 holds. If the Bohr-Fourier spectrum of g is empty, then 1 f (n)g(n) → 0 (x →∞) x n≤x for each q-multiplicative function f of modulus 1. (183)

~~413 CHAPTER 4~~

~~11 The G-ary digital expansion of a number~~

Let G = (G j ) j≥0 be a strictly increasing sequence of integers with G0 = 1. Then every nonnegative integer n has a unique proper G-ary digital expansion n = a j (n)G j j≥0 with integer digits a j (n) ≥ 0 provided that a j (n)G j < Gk for all k ≥ 0. j

The distribution of sG (n) has been studied by M. Drmota and J. Gajdosik [82]. Let (X N ) be a sequence of discrete random variables X N , (N ≥ 0) deﬁned by card{n < N : s (n) = k} Pr[X = k] = G N N

~~The expected value and variance of X N are given by 1 1 2 EX N = sG (n), VXn = (sG (n) − EXn) N n~~

1) There exist nonnegative integers gi (1 ≤ i ≤ r) such that for j ≥ r, r G j = gi G j−i i=1 ( ) 2) gcd{i ≥ 1: gi = 0}=1 184 3) For all j > r and 1 ≤ k < r we have r G j−k ≥ gi G j−i i=k+1 We note that the above assumptions imply that the characteristic polynomial r r r−i P(u) = u − gi u has a unique root α of maximal modulus which is real i=1

414 SPECIAL ARITHMETIC FUNCTIONS and positive, and that j G j ∼ Cα as j →∞ where C > 0isaconstant. j When g1 ≥ g2 ≥ ··· ≥ gr ≥ 0 and G j = g j G j−1 + 1 for j < r, then all i=1 assumptions are satisfied. Furthermore, P(u) is irreducible and α is a Pisot number, see [159]. The fact that α is a Pisot number, is essentially due to A. Brauer [38]. For Pisot (or Pisot-Vijayaraghavan) and Salem numbers, see e.g. the monograph [24] by M. J. Bertin, et al. For more recent results, see D. W. Boyd [35], D. W. Boyd and R. D. Mauldin [37], or M. J. Bertin and D. W. Boyd [23]. For connections to Fourier analysis, see R. Salem [297], and to Harmonic analysis, see Y. Meyer [256]. For infinite recurrences our starting point is Parry’s α-expansion of 1 (see [270]) g g g 1 = 1 + 2 + 3 + ... (185) α α2 α3 where α>1isareal number and gi (i ≥ 1) are positive integers. (In the case of ambiguity we take the infinite representation of 1.) The sequence G = (G j ) j≥0 is now defined by

~~j G0 = 1, G j = g j G j−1 + 1 ( j ≥ 1)(186) i=1~~

If we further set i j A(u) = gi u , G(u) = G j u , i≥1 j≥0 then 1 G(u) = (1 − u)(1 − A(u)) 1 z = > and it follows that 0 α 0isthe only singularity on the circle of convergence |z|=z0,which is a simple pole. Hence

~~ j G j ∼ C α as j →∞ and so we are in similar situation as above. The following theorem is a collection of results from [272], [158], [159] etc. (see [82]):~~

~~415 CHAPTER 4~~

Suppose that G = (G j ) satisfies a finite or infinite linear recurrence of the above types (see (184), (185), (186)). Set − gj 1 +···+ G(z, u) = zl zg1 g j−1 u j j≥1 l=0 and let 1/α(z) denote the analytic solution u = 1/α(z) of the equation G(u, z) = 1 for z in a sufficiently small (complex) neighbourhood of z0 = 1 such that α(1) = α. Then log N EX = µ + O(1), (187) N log α and log N VX = σ 2 + O(1), (188) N log α α(1) α(1) µ = σ 2 = + µ − µ2 where α and α . For the asymptotic normality in the weak sense we have: 1 |card{n < N : S (N)

~~416 SPECIAL ARITHMETIC FUNCTIONS~~

12 The sum-of-digits function for negative integer bases The number systems with negative integer bases were ﬁrst introduced by V. Grunwald¨ in 1885, in a little known journal [161]. Later the idea was rediscovered by A. J. Kempner in 1936, Z. Pawlak and A. Wakulicz in 1957, I. Wadel in 1957, D. E. Knuth in 1955 (see the book [224] by D. E. Knuth). Knuth studied also numerical systems with complex bases. It is easy to prove that for an integer q ≥ 2, every n ∈ Z has a unique representation of the form

2 h n = c0 + c1(−q) + c2(−q) + ···+ch(−q) (192) with ci ∈{0, 1,...,q − 1},0≤ i ≤ h, and c j = 0 for h = 0. Expression (192) is called (−q)-adic representation of n. The sum of digits function of n now is deﬁned by

~~s−q (n) = c0 + c1 + ···+ch~~

~~For the mean value of s−q (n): ( ) = ( − ) + ( ), ( ) s−q n q 1 N logq N NF logq2 N 193 |n|~~

~~1 (q − 1)(q2 − 4q + 7) r (q) = · 1 6 (q + 1)~~

~~417 CHAPTER 4~~

1 (q − 1)(q6 − 11q5 + 82q4 − 322q3 + 577q2 − 371q + 76) r (q) = · , 2 180 (q + 1)3 1 r (q) 1 ψ(x) = √ · H(x) + 2 − √ , π ( ) 3 2 π ( ) 2 r1 q 2 πr1(q) 3Q r1 q and Q = 2ifq is even; and = 1, if q is odd. As a corollary one obtains that " − ( ) q 1 r1 q 1/2 {s− (n), s− (−n)}= N N − N( N) + min q q logq π logq n

13 The sum-of-digits function in algebraic number ﬁelds H. G. Senge and E. G. Straus [305] in 1973 have discovered the following interesting fact. If (q1, q2) = 1 and c > 0isaconstant, then there are only ﬁnitely many n ≥ 0such that ( ) ≤ ( ) ≤ ( ) sq1 n c and sq2 n c 200 This result was later generalized and sharpened by C. L. Stewart [318], H. P. Schlickewei [302], [303], and A. Petho¨ and R. F. Tichy [272], [274]. The proofs use A. Baker’s method on linear forms of logarithms and the p-adic version of W. Schmidt’s subspace theorem by Schlickewei applied to S-unit equations. In [274] the sum of digit function with respect to canonical number system in an algebraic

418 SPECIAL ARITHMETIC FUNCTIONS number ﬁeld is considered. In the case of Gaussian integers, these system have been studied in section 9.Inthe most general case, these were introduced by Katai´ and Kornyei¨ [210]. Let K be a number ﬁeld of degree [K ; Q] = n, and let Z K denote its ring of integers. Then it is well known (see e.g. [31]) that there exist β1,...,βk ∈ K such that any γ ∈ Z K has a representation

~~γ = b1β1 + ···+bnβn (bi ∈ Z)~~

The vector (β1,...,βn) is called an integral basis of Z K .Forα ∈ Z K we denote the norm of α by N(α).Ifwehaveα ∈ Z K , and N ={0,...,|N(α)|−1}, then the pair (α, N ) is called a canonical number system if any γ ∈ Z K has a unique ﬁnite digital representation of the form

~~h γ = c0 + c1α + ···+chα , c j ∈ N ( j = 0,...,h)(201)~~

~~Here α is called the base of the number system, N its set of digits, while (201) is called the α-adic representation of γ . The length of γ is deﬁned by~~

~~Lα(γ ) = h~~

In case of Z[i] the bases have been characterized by Katai´ and Szabo´ [212], in the more general case of quadratic number fields by Katai´ and Kovacs´ [211]. Any quadratic number field contains bases that form canonical number systems. We note that this is not true in number fields of higher degree. B. Kovacs´ [234] in 1981 gave a criterion for a number field to contain canonical number systems: In Z K there exists a canonical number system iff there exists an α ∈ Z K ,such that {1,α,...,αn−1} is an integral bases. (202) The complete characterization was established in 1991 by Kovacs´ and Petho¨ [235]: There exist α1,...,αt ∈ Z K ; n1,...,nt ∈ Z,andN1,...,Nt finite subsets of Z, which are effectively computable, such that (α, N ) is a canonical number system in Z K if and only if α = αi − h for some integers i, h with 1 ≤ i ≤ t and either h ≥ ni or h ∈ Ni .(203) By using this result, and an algorithm due to K. Gyory¨ [166], Kovacs´ and Petho¨ were able to develop an algorithm for finding the canonical number systems of the ring of integers of a number field. The sum of digits function sb(γ ) of a number γ with b-adic representation

~~h γ = c0 + c1b + ···+chb will be sb(γ ) = c0 + ···+ch.~~

~~419 CHAPTER 4~~

A general result for the summatory function of sb has been obtained by J. M. Thuswaldner [338] in 1998. Let θ be the primitive element of the number ﬁeld K , and let θ1,...,θ2 be its n−1 conjugates. Then any α ∈ K has a representation γ = r1 + r2θ + ···+rnθ , with ri ∈ Q (i = 1, n).Deﬁne the mappings

σ1,...,σs,σs+1, σ s+1,...,σs+t , σ s+t which map K = Q(θ) onto the equivalent ﬁeld Q(θi ) and fulﬁll σi |Q = idQ.Here, s is the number of real roots, t is the number of pairs of conjugate complex roots of the minimal polynomial of θ, hence the degree of K is n = s + 2t. The pair (s, t) is called the signature of K .With the help of these mappings we can transform each element γ ∈ K into the vector

x(γ ) = (σ1(γ ), ...,σs(γ ), σs+1(γ ), ...,σs+t (γ )) The vectors are elements of a vector space of dimension s + 2t over R. One can show that the elements of Z K form a lattice in this vector space. The fundamental domain of this lattice, and any translate of it, is a parallelotope. If we assume that Z K contains a canonical number system, then by theorem (202) there exists an integral basis of the form {1,α,...,αn−1}.Sothe vectors x(1), . ..,x(αn−1) generate the lattice induced by Z K .Byusing Gram’s determinant, the volume of the fundamental domain will be: I (α) = det(x(αi ), x(α j )), i, j = 0, n − 1 We consider regions of the following form:

~~|z1| < l1,...,|zs| < ls, zs+1zs+1 < ls+1,...,zs+t zs+t < ls+t (204) where z1,...,zs, zs+1, zs+1,...,zs+t , zs+t are the conjugates of z ∈ K , and l1,...,ls+t are certain numbers. Let~~

D(l1,...,ls; ls+1,...,ls+t ) denote the set of all z ∈ K satisfying (204). The following theorem holds true: Let K be a number ﬁeld of degree n and signature (s, t), and Z K its ring of integers with integral bases {1,α,...,αn−1}. Let b be the base of a canonical number system, and let M denote the absolute value of the norm of b over Q. Let b1,...,bs; bs+1, bs+1,...,bs+t , bs+t be the conjugates of b; let 1 < x < |b1| (and 2 1 < x < |b1| if s = 0), and

~~|bi |−1 |b1|−|bi | x1 = x, xi (x) = ai x + ci with ai = , ci = ,(i = 2,...,s) |b1|−1 |b1|−1~~

~~420 SPECIAL ARITHMETIC FUNCTIONS and~~

~~2 2 |bi | − 1 |b1|−|bi | xi (x) = ai x + ci with ai = , ci = (i = s + 1,...,s + t) |b1|1 |b1|−1 Furthermore, put~~

~~k k 2k 2k D(b, k, x) = D(|b1| x1(x), ...,|bs| xs(x);|bs+1| xs+1(x), ...,|bs+t | xs+t (x))~~

~~k and N = M x1 ...xs+t . Then we have~~

s t 2 π M − 1 n−1 s (z) = · N log N + Nφ(log N)+ O(N n log N)(205) b I (α) M M M z∈D(b,k,x) 2 where the sum runs over all algebraic integers in D(b, k, x),andφ is a continuous periodic function of period 1. The special case of K = Q[i], b =−n ± i, s = 0, t = 1, n = 2 contains a result by P. J. Grabner, P. Kirschenhofer and H. Prodinger [152]: √ ( ) = π + φ( ) + ( )() sb z 2 N logN(b) N N logN(b) N O N logN(b) N 206 |z|2

14 The symmetric signed digital expansion If we write the word 1111010000100100000 (which is 106 in the binary system), the sublock 100 is contained three times. A generalization of the number of occurences (i.e. frequency) of a digit, one can consider the frequency of a block, i.e. 1 (number of occurences of w) N 0≤n

~~421 CHAPTER 4~~

~~C. Heuberger and H.Prodinger [181] have considered a symmetric system with q q an even base q, and digits a ∈ − ,..., ,and for which j 2 2 j n = a j q j≥0~~

Such a system is a priori redundant because of the existence of both ±q/2butcan q be made unique by the condition that |a |=q/2 implies 0 ≤ sign(a )a + < . j j j 1 2 This expansion is called symmetric signed digit expansion by P. J. Grabner, C. Heuberger and H. Prodinger [151], who study the subblock counting problem in such cases. l If a block b = (bs,...,b0) is given, we denote its value by v(b) = bkq .We l use the Iverson notation [P], i.e. [P] = 1ifproperty P is true, and = 0, otherwise. Then the number of subblock occurrences of b will be sb(n) = [(ak+r−1(n),...,ak(n)) = b] k≥0

~~A block b is called admissible,itifrepresents the number v(b) in the symmetric signed digit expansion. Let Sb(N) = sb(n). Then the following theorem holds n~~

~~422 SPECIAL ARITHMETIC FUNCTIONS~~

R (b ) h (b) = log [v(b)<0] + q−r v(b) + min 0 − 0 q qr (q + 1) R (b ) − log [v(b)<0] + q−r v(b) + max 0 − q qr (q + 1) Q(b ) 1 1 1 1 − 0 r + + − + qr (q + 1) 2 log q q + 1 qr−1(q + 1) and where $ % q q R (η) =− − (η − 1)mod q ≥ , min 2 2 $ % q q R (η) = + ηmod q < max 2 2

The function Hb(x) is a periodic continuous function of period 1, and mean 0. (ζ(s, x) denotes the Hurwitz zeta function.) The case r = 1was discussed in [181], and that of q = 2, r = 1 (via Dirichlet series and Mellin-Perron summation formula) in [340]. The following central limit theorem is valid, too: ( ) 1 < ( )< Q b0 + = lim card n N : sb n logq N x Vb logq N N→∞ N qr (q + 1) x 1 2 = √ e−t /2dt (208) 2π −∞ where r−1 Q(b0) 1 V = 1 + 2 [b = b ,...,b − = b − − ] + b r ( + ) i 0 r 1 r 1 i i q q 1 i=1 q q Q(b ) Q(b ) +2 1 − 0 − 2(r − 1) 0 qr (q + 1) q + 1 qr (q + 1)

15 Inﬁnite sums and products involving the sum-of-digits function There exist many interesting inﬁnite sums or products involving the sum of digit or related functions. The sum s(n) of decimal digits of n has the generating function

~~∞ ∞ n n n xs(n)tn = (1 + xt10 + x2t2·10 + ···+x9t9·10 )(209) n=0 n=0~~

~~423 CHAPTER 4 for |t| < 1, from which we can deduce that~~

~~∞ ( ) s n = 10 ( ) ( + ) log 10 210 n=1 n n 1 9~~

Foraproof of (210) see e.g. J. M. Borwein and P. B. Borwein [32]. More generally, ∞ s (n) q q = q ( ) ( + ) − log 211 n=1 n n 1 q 1 For q = 2 there exist sums as (see J. P. Allouche [4] and J. P. Allouche and J. Shallit [10]): ∞ 2n + 1 π 2 s (n) = ,(212) 2 2( + )2 n=1 n n 1 9

∞ 8n3 + 4n2 + n − 1 17 (s (n))2 = + log 2 (213) 2 ( 2 − )( 2 − ) n=1 4n n 1 4n 1 24 For ∞ 1 1 2k s (n) − = (1 − 21−k)ζ(k)(214) 2 k ( + )k k − n=1 n n 1 2 1 see [32]. Let D(n) = ∗i,where the star means that the product is taken over those i where the ith binary digit of n is non-zero. Then ∞ ( ) π/ −π/ (−1)s2 n e 2 − e 2 = 2 (215) s2(n)( ( ))4 π 2 n=0 16 D n

~~∗ Let t(n) = i, where the sum is taken over the non-zero digits of n base 2 (e.g. t(10112) = 4 + 0 + 2 + 1 = 7). Then~~

~~∞ ( ) ∞ (−1)s2 n 2 · (−1)n = (216) ( ) + 2 + + n=0 t n 1 n=−∞ 3n n 2~~

~~The inﬁnite product~~

~~∞ (− )s (n) 2n + 1 1 2 = −1/2 ( ) + 2 217 n=0 2n 2 is due to D. R. Woods and D. Robbins [355]. J. O Shallit [306] generalized it to sq (n).~~

~~424 SPECIAL ARITHMETIC FUNCTIONS~~

Let q ≥ 2, 1 ≤ j ≤ q − 1 and cn, dn be the unique integers satisfying qn ≤ cn, dn < q(n + 1) and sq (cn) ≡ j − 1 (mod q), sq (dn) ≡ j (mod q). Then ∞ 1 + c n = q−1/q ( ) + 217 n=0 1 dn For q = 2 one gets (217). Another generalization is the following: Let q ≥ 2 even and let an, bn be such that 2n ≤ an, bn < 2(n + 1), and sq (an) ≡ 0 (mod 2), ( ) ≡ ( ) sq bn 1 mod 2 . Then √ ∞ + q 1 an = ( ) + 217 n=0 1 bn q Other general inﬁnite products are treated in Allouche et al. [7]. Forexample,

~~∞ ( )·(− )s (n) ∞ −( / )s (n) 2n + 1 s2 n 1 2 2n + 1 3 2 2 = 1/4, = −2/5 + 2 + 2 n=0 2n 2 n=0 2n 2~~

~~Let qw(n) count the number of (possibly overlapping) occurrences of the block w in the binary expansion of n.Then~~

~~( ) √ ∞ (−1)a0 n 2n = 2, + n=1 2n 1 2~~

( ) √ ∞ ( + )( + )( + )( + ) (−1)a1010 n 4n 2 8n 7 8n 3 16n 10 = 2, ( + )( + )( + )( + ) n=0 4n 3 8n 6 8n 2 16n 11 2 see J.-P. Allouche and J. O. Shallit [11]. More generally, there exists an effectively computable rational function bw(n) such that aw(n) 1 (bw(n))X = log2 − n≥0 X 1 for all X = 1, with |X|≤1. See also J.-P. Allouche, P. Hajnal and J. O. Shallit [8]. Let a(n) = number of odd digits in odd places in the decimal expansion of n, b(n) = number of odd digits in n (e.g. a(901) = 2, a(811) = 1, b(901) = 2, b(811) = 2). Then ∞ ( n) ∞ ( n) a 2 = 1 , b 2 = 1, n n n=1 2 99 n=1 2 9 (218) ∞ 1 1 25π 2 b(n) − = 2 ( + )2 n=1 n n 1 297

~~425 CHAPTER 4~~

Let u(n) be the ”reﬂection” of n through the decimal point (e.g. e(123) = 0.321, e(90140) = 0.04109). Then ∞ ( ) u n = 10 ( ) ( + ) log 10 219 n=1 n n 1 99 For the sums (215), (216), (218), (219), see [32]. Let g(n) be the number of digits ≥ 5inn. Then

∞ ( n) g 2 = 2,() n 220 n=0 2 9 see E. Levine [242] and D. Bowman and T. White [34]. Let e(n) denote the number of digit blocks of 11 in the binary expansion of n. Then ∞ ( ) e n = 3 − 1π() ( + ) log 2 221 n=1 n n 1 2 4 see Allouche [4]. Let v(n) be the number of terminating nines when the ﬁrst n integers are written in base 10. Then v(n) = [n · 10−i ], and the set of values taken by the function v i≥1 is exactly the set of integers k written as

K i k = ((10 − 1)/9)ni , i=1 where 0 ≤ ni < 10 for all i ∈{1,...,K }. R. K. Kennedy et al. [216] proved that 1 9 lim card{v(n) :1≤ n ≤ M}= M→∞ M 10 By using Roth’s theorem of Diophantine approximation theory, it can be shown that (see [32]) ∞ ( ) . ∞ ( n) . a n = 10 e 2 = 3166 ( ) n and n are transcendental 222 n=1 10 9 n=1 2 3069 We note that by considering formal power series, J. P. Allouche [5] proved that if R is a polynomial in Q[X] with R(N) ⊂ N, then the formal power series ∞ n sp(R(n))X n=0

~~426 SPECIAL ARITHMETIC FUNCTIONS~~

is algebraic over Fp(X) (where Fp is the field of p elements (p=prime), and Fp(X) is the field of rational fractions with coefficients in Fp)ifand only if the degree of R is less than or equal to 1. (223) (k) If sp denotes the kth iterate of sp, then the formal power series ∞ (k)( ) n sp n X n=0 is algebraic over Fp(X) if and only if k = 0ork = 1. (224)

16 Miscellaneous results on digital expansions We now study some miscellaneous facts related to the digital expansions of numbers. First we consider certain iteration problems. Let f : N → N be an arithmetic function, and consider the f -iterative sequence

~~(i) n0 = n, n1 = f (n0), . ..,ni = f (ni−1) = f (n0),...~~

This sequence is called periodic, when there are integers i0 ≥ 0, and l ≥ 1such ≥ = { ,..., } that, for all i i0, ni+l ni . The set ni0 ni0+l−1 is called a cycle of f and l the length of the cycle. The function f is called periodic,if f has ﬁnitely many cycles. R. E. Burkard [45] gave a characterization of periodicity: f is periodic if and only if there exists an N ∈ N and for all n > N a k0 = k0(n) such that for all n > N, f k0 (n)

~~The function f is called uniformly periodic if for all n > N there exists k0 ≥ 1, independent of n, such that f k0 (n)~~

0, m − 1), 1 ≤ am ≤ q −1betheq-adic expansion of n. Let P : {0, 1,...,q −1}→ m N be an arbitrary function, and deﬁne f (n) = P(a j ).(226) j=0 In 1960, B. M. Stewart [317] showed that there exists an integer N = N(P) such that f (N) ≥ N and f (n) N (227) Therefore this function is uniformly periodic. Stewart gave also al algorithm for the evaluation of N in (226), which is of crucial importance for the determination of all cycles of f .

~~427 CHAPTER 4~~

We note that for special choices of P, result (226) has been rediscovered by many writers after Stewart. For a history of these facts, see also J. Sandor´ [300]. For P(a) = at (t ∈ N,ﬁxed), in the case of q = 10, t = 2 all cycles were computed by A. Porges, by K. Iseki´ for t = 3, ... ,byK.Iseki´ and I. Takada for t = 9. N. Yoshigahara obtained all cycles for q = 100 and t = 2. For general base q,H.Hasse and G. Prichett investigated the cycles in the case t = 2. For References, and related results, see H. J. J. te Riele [286]. For P(a) = a!, q = 10, G. D. Poole found all cycles of length 1, i.e. {1}, {2}, {145}, {40585}.P.Kiss found all other cycles (i.e. two of length 2 and one of length 3). q For P(q) = (binomial coefﬁcient), and q = 10, P. Kiss found all cycles a (one of length 1 and one of length 2). q For P(a) = a! ,andq = 10, P. Kiss found also all cycles (one each of length a 2, 4, 26 and 39). See [286]. A similar function is m f (n) = P(a j ), (228) j=0 where P : {0, 1,...,q − 1}→{0}∪N.B.M.Stewart [317] showed that there exists an integer N ≥ 1for which f (N) ≥ N and f (n) N, only in the case that

~~1 ≤ M ≤ M < B and P(a) ≥ a for at least one a ∈{0,...,q − 1} (229)~~

Here M = max P(a) and M = max{P(0), M}. 1≤a≤q−1 Moreover, he showed that in all other cases either N = 0orN doesn’t exist. For the particular case of P(a) = a + t (t ≥ 0ﬁxed), q = 10, S. S. Wagstaff, Jr. made a through computational study of the ten cases 0 ≤ t ≤ 9 (for t ≥ 10 we have f (n)>n for all n). He showed that for t = 1 and n > 18, either f (n)

~~428 SPECIAL ARITHMETIC FUNCTIONS~~

The function f (n) = n − n (231) where n is the integer formed by arranging the base q digits of n in descending order, and n by arranging them in ascending order, was first studied by D. R. Kaprekar [203], [204]. For q = 10 he discovered that when we start with a four-digit-number, not all digits being identical, then within 8 steps the cycle {6174} is reached. We note that the following convention is generally adopted: if n is a k-digit number and f (n) has k − 1 digits, then a leading zero is added to f (n).(E.g. (q = 10) n = 1121, f (n) = 2111 − 1112 = 0999, f (2)(n) = 9990 − 0999 = 8991). When we neglect numbers all whose digits are equal, this means that iterating f on k-digit numbers leads always to k-digit numbers. J. H. Jordan studied cycles of length 1 for any base q;forb = 2, 3,...,12, C. W. Trigg listed all cycles for two-digit and five-digit numbers. H. Hasse presented a detailed study of the problem for two-digit numbers, for general q.G.Prichett gave acomplete characterization of all cycles in the case of five q-adic integers. A. L. Ludington in 1979 showed that for any fixed base q there exists an r0 ∈ N such that for all r-digit numbers with r > r0 there is no cycle of length 1, while G. D. Prichett, A. L. Ludington and J. F. Lapenta [279] showed that for q = 10 the only cycles of length 1 are 495 (for 3-digit numbers) and 6174 (for 4-digit numbers). For a variation of the two-digit Kaprekar routine, see A. L. Young [358]. Finally, let f (n) = (n + t)∗ (t ∈ N)(232) where t is fixed and m∗ is the number obtained by reversing the digits of m. This function is not periodic (put t = 10), since the number of cycles is infinite, as was shown by Gorzkowski (see [286]) by the followimg example: 2k+3 k+1 Let n0 = 10 + 10 + 1, and consider the iteration of f at n0. Then this f - iterative sequence has a period length 36 · 10k (for any fixed k). W. Sierpinski [311] gave results, by showing that f is periodic for t = 1, 2, 3, 4, 7, 8, 9, 11. For t = 6 the question is open (q = 10). The iteration of s(n) has been studied also by K. T. Atanassov [13] (this is a par- t ticular case of (sq (n)) studied above). For an application to ”staggered sums”, see A. G. Shannon and A. F. Horadam [307]. U. Balakrishnan and B. Sury [14] considered f (n) = s(d)(233) d|n and proved that for all n > 1 there is an N = N(n) such that

~~f (N)(n) = 15 (234)~~

~~429 CHAPTER 4~~

For the iteration of f (n) = σ(n), σ(n) − n see Chapter 1, while for the iteration of f (n) = ϕ(n) and related functions, see Chapter 3. Two properties of the digits of Fibonacci numbers are the following. Let Am(n) be the number of Fibonacci numbers Fn with n ≤ N and whose ﬁrst digit in base q is m (where 1 ≤ m ≤ q − 1). Then L. C. Washington [348] proved that the Benford law holds true: + 1 ( ) = m 1 ( ) lim Am N logq 235 N→∞ N m

Let now x(k) be the number of indices n such that Fn has k decimal digits. Then x(k) ∈{4, 5} for k ≥ 2. Let A(N) be the number of k with 2 ≤ k ≤ N such that x(k) = 5. Then A. Guthmann [165] proves that

A(N) = αN + o(1) as N →∞,(236) √ 1 + 5 where α = log 10/ log − 4 = 0.78497 ... The proof is based on Baker’s 2 method on linear forms in logarithms. There exist result also for base q palindromes. m If n = a0 + a1q + ··· +amq with ai ∈{0,...,q − 1}, am = 0, then n is called a base q palindrome (not confusing with the notion of a palindromic sequence x0,...,xn for which xk = xn−k for 0 ≤ k ≤ n)ifak = am−k for all k = 0, 1,...,m (thus the sequence of digits is a palindromic sequence). For some properties of base q palindromes, see e.g. I. Korec [233], M. Harminic and R. Sotak´ [178]. In 1998 F. Luca [244] has considered the palindromes in a row of the Pascal triangle. He posed the following questions: 1. Given a positive integer q > 1, ﬁnd the values of n for which, if one writes each binomial coefﬁcient belonging to the n + 1th row of the Pascal triangle in base q, the sequence resulting by concatenating all these blocks of digits (in the natural order), is a palindrome. By abuse of terminology, we will refer to the n + 1th row of the Pascal triangle for such n as a base q-palindrome. In fact the question is equivalent to the following one: 2. What are the values of n for which n (237) k is a base q-palindrome for all k = 0, 1,...,n? In [244] it is shown that if q = 10, then n ≤ 4. (238)

~~430 SPECIAL ARITHMETIC FUNCTIONS~~

Further, in [245] Luca showed that for any q > 1 there exist only finitely many n’s satisfying the requirement of question 2. More precisely: (239) 1) There exists a constant N(q) such that, if n has the property that all binomial coefficients of list (217) are base q palindromes, then n ≤ N(q). 2) N(2) = 3, N(4) = 5, N(6) = 7. Moreover, if q ∈{2, 4, 6}, then N(q) 1, with index k,if a2 ≡ a (mod qk)(240) Forexample, 376 is automorphic in the scale 10 with index 3, since 3762 ≡ 141376. He proves the following results: There are automorphic numbers in the scale q, with all indices, if and only if n is expressible as a product of two relatively prime factors. (241) Let q be the product of relatively prime factors u,v, and let k ≥ N be chosen so that uk ≡ 1 (mod vN ). Then the remainder when uk is divided by q N is an automorphic number in the scale q with index N.(242) Finally, Goodstein proved that if q is a product of relatively prime factors u,v and k is chosen so that uk ≡ 1 (mod v), − then the remainder when ukvN 1 is divided by q N is an automorphic number in the scale q with index N.(243) In fact, theorem (243) remains true, if one replaces u by any w divisible by u and prime to v. A. R. Pargeter [269] had identified four automorphic numbers in scale 10. In 1993 T. Gagen [136] defined n-automorphic numbers (in scale 10), with index k by an ≡ a (mod 10k), (244) and examined the case n = 5indetail. For k = 2a · 5b + 1heobtained some more general results. In 1991 E. B. Knisch [239] introduced modular idempotent numbers a,having the property a2 ≡ a (mod 10l(n)), (245) where l(n) denotes the number of decimal digits of n.For example, 52 = 25, 762 = 5776 so both 5 and 76 are modular idempotents, but 57762 = 3362176 shows that 5776 is not of this type.

~~431 CHAPTER 4~~

= · j−1 ( ≤ ≤ ) τ( , ) = k · j−1 ( ≥ ) Let n a j 10 0 a j 9 , and put n k j=1 a j 10 k 1 , j≥1 2 2 and γ(n) = min{τ(n , k) : τ(n , k)>n, k ≥ 1}. Let P1 = 5, Q1 = 6, and deﬁne for n ≥ 2 Pn, Qn recursively by

~~k−1 Pn+1 = γ(Pn), Qn+1 = γ(Qn) + (10 − 2sk) · 10 ,~~

k j−1 where γ(Qn) = s j · 10 , sk = 0. j=1 Then for all n ≥ 1, Pn and Qn are modular idempotents. In fact, m ≥ 2ismodular idempotent if and only if m = Pn or m = Qn for some n ≥ 1. (246) Finally, we mention that there exists an extensive literature on other expansions (which we do not consider here) and various digit functions and various property (e.g. ergodic) related e.g. to continued powers and roots (see e.g. D. J. Jones [197], [198], G. Polya´ and G. Szego[¨ 276]), continued fractions (see e.g. W. B. Jones and W. J. Thron [199], I. J. Good [147], D. Hensley [179], O. Jenkinson and M. Pollicott [194], beta-expansions (see e.g. A. Renyi´ [285] and W. Parry [270], D. W. Boyd [36], F. Blanchard [30], V. A. Rohlin [287], J. B. Pesin [273]), f -expansions and Bolyai- Renyi´ expansions (see e.g. A. Renyi´ [285], O. Jenkinson and M. Pollicott [195]), etc. Forvarious representations of real numbers, see J. Galambos [139].

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~~[339] J. M. Thuswaldner, Fractal dimension of sets induced by bases of imaginary quadratic ﬁelds,Math. Slovaca, 48(1998), no. 4, 365-371.~~

~~[340] J. M. Thuswaldner, Summatory functions of digital sums occuring in cryptog- raphy,Period. Math. Hungar., 38(1999), 111-130.~~

~~455 CHAPTER 4~~

~~[341] J. M. Thuswaldner and R. F. Tichy, An Erdos-Kac¨ theorem for systems of q-additive functions, Indag. Math. (N.S.), 11(2000), no. 2, 283-291.~~

~~[342] J. M. Thuswaldner and R. F. Tichy, Waring’s problem with digital restrictions, Preprint.~~

~~[343] R. F. Tichy and G. Turnwald, On the discrepancy of some special sequences, J. Number Theory, 26(1986), 351-366.~~

~~[344] J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., 41(1968), 21-25.~~

~~[345] M. D. Vose, Integers with consecutive divisors in small ratio,J.Number The- ory, 19(1984), 233-238.~~

~~[346] M. D. Vose, The distribution of divisors of N!, Acta Arith., 50(1988), 203-209.~~

~~[347] R. Warlimont, Naturliche¨ Zahlen mit primen Quersumme,Arch. Math. (Basel), 45(1985), 144-149.~~

~~[348] L. C. Washington, Benford’s law for Fibonacci and Lucas numbers, Fib. Quart., 19(1981), 175-177.~~

~~[349] A. Weingartner, Integers with dense divisors,toappear J. Number Theory.~~

~~[350] A. Weingartner, Integers with dense divisors 2, submitted.~~

~~[351] E. Weisstein, http://mathworld.wolfram.com/HarshadNumber.html~~

~~[352] E. Weisstein, http://mathworld.wolfram.com/Sum-ProductNumber.html~~

~~[353] E. Weisstein, http://mathworld.wolfram.com/SmithNumber.html~~

~~[354] A. Wilansky, Smith numbers,Two Year College Math. J., 13(1982), 21.~~

~~[355] D. R. Woods, Problem E2692, Amer. Math. Monthly, 85(1978), no. 1, p. 48, solution by D. Robbins, same journal, 86(1979), 394-5.~~

~~[356] S. Yates, Special sets of Smith numbers,Math. Mag., 59(1986), 293-296.~~

~~[357] S. Yates, Smoth numbers congruent to 4 (mod 9),J.Recr. Math., 19(1987), 139-141.~~

~~[358] A. L. Young, A variation of the two-digit Kaprekar routine,Fib. Quart., 31(1993), no. 2, 138-145.~~

~~456 SPECIAL ARITHMETIC FUNCTIONS~~

~~[359] U. Zannier, On the distribution of self-numbers, Proc. Amer. Math. Soc., 85(1982), 10-14.~~

~~[360] E. Zeckendorf, Representation´ des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liege,` 41(1972), 179-182.~~

~~457 Chapter 5~~

~~STIRLING, BELL, BERNOULLI, EULER AND EULERIAN NUMBERS~~

~~5.1 Introduction~~

This Chapter is divided into two major parts: Stirling and Bell numbers, and Bernoulli, Euler and Eulerian numbers. These classical topics occur in practically every field of mathematics, in particular in combinatorial theory, finite difference calculus, numerical analysis, number theory, and probability theory. Our aim is to study the many aspects of these numbers, and to point out important connections or applications in number theory and related fields. In fact this Chapter underlines the many connections and aspects of certain parts of Discrete mathematics to almost the entire mathematics.

~~5.2 Stirling and Bell numbers~~

1 Stirling numbers of both kinds, Lah numbers Let S(n, k) denote the number of partitions of a set with n elements into k nonempty blocks. Then S(n, k)>0for1≤ k ≤ n, and S(n, k) = 0 for 1 ≤ n < k. Put S(0, 0) = 1andS(0, k) = 0 for k ≥ 1, S(n, 0) = 0 for n ≥ 1. These numbers were introduced by J. Stirling in his book ”Methodus Differ- entialis” [424] from 1730 (see also O. Schlomilch¨ [394], J. F. Steffensen [423],

~~459 CHAPTER 5~~

N. Nielsen [333], G. Boole [56], Ch. Jordan [237] for early references, and various other notations), and are called now as the Stirling numbers of the second kind. These numbers are treated also e.g. in J. Riordan [365], [367], L. Comtet [116], P. A. Mac Mahon [295], N. E. Norlund¨ [338], L. M. Milne-Thomson [310], M. Aigner [15], P. J. Davis [131], M. Abramovicz and I. A. Stegun [1], A. O. Gelfond [180], F. W. J. Olver [343], I. Tomescu [448], etc. See also D. S. Mitrinovic´ and R. S. Mitri- novic´ [311], where various tables of Stirling numbers can be found, too. The Stirling numbers of the second kind S(n, k) satisfy the recurrence relation

S(n, k) = S(n − 1, k − 1) + kS(n − 1, k)(n, k ≥ 1)(1) and, if we denote by (x)n = x(x − 1)...(x − n + 1) the falling factorial power, then n n x = S(n, k) · (x)k (2) k=0 where (x)0 = 1bydeﬁnition. This gives a ”horizontal” generating function of S(n, k). The ”vertical” generating function is given by tn 1 S(n, k) = (et − 1)k (k ≥ 0)(3) n≥0 n! k! (Here one can take n ≥ k in place of n ≥ 0, since S(n, k) = 0 for n < k). Similarly, the ”rational” generating function is 1 S(n, k)un−k = (k ≥ )() ( − )( − )...( − ) 1 4 n≥0 1 u 1 2u 1 ku In fact, S(n, k) can be explicitly calculated with the aid of binomial coefﬁcients: 1 k k S(n, k) = (−1)i (k − i)n (5) k! i=0 i and have the following vertical, resp. horizontal recurrences (see e.g. [116])

~~− n 1 n − 1 S(n, k) = S(l, k − 1), (6) l=k−1 l~~

~~n S(n, k) = S(l − 1, k − 1)kn−l ,(7) l=k n−k i S(n, k) = (−1) k + 1i S(n + 1, k + i + 1), (8) i=0~~

~~460 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

where xn = x(x + 1)...(x + n − 1) (with x0 = 1) denotes the rising factorial power. The number of all partitions of a set with n elements is n B(n) = S(n, k), (9) k=1 called also a Bell number (or exponential number). These numbers verify the recurrence n n B(n + 1) = Bk (k ≥ 0), (10) k=0 k where B(0) = 1bydeﬁnition. The generating function of B(n) is given by tn B(n) = exp(et − 1)(11) n≥0 n! The numbers B(n) can be represented also as the sum of a convergent series (Dobinski’s formula) 1 kn 1 1n 2n 3n B(n) = = + + + ... ,(12) e k≥0 k! e 1! 2! 3! see also G. Polya´ and G. Szego¨ [352] for these basic properties. The notation B(n) for Bell numbers should not be confused with Bn, the nth Bernoulli number, which can be obtained (formally) from

n (1 + B) − Bn = 0,(13) where B0 = 1. This gives 1 + 2B1 = 0, 1 + 3B1 + 3B2 = 0, 1 + 4B1 + 6B2 + 4B3 = 1 1 1 1 0,..., implying that B =− , B = , B =− , B = ,... and B + = 0 1 2 2 6 4 30 6 42 2n 1 for n ≥ 1. The generating function of (Bn) is tn t B = (14) n t − n≥0 n! e 1 The Bernoulli numbers can be expressed by the Stirling numbers of the second kind as follows (see e.g. [237]):

~~+ n 1 (−1)kk! B = S(n, k)() n + 15 k=1 k 1~~

~~461 CHAPTER 5~~

For Bernoulli numbers, very important in many parts of Mathematics, see e.g. N. Nielsen [334], L. Saalschutz¨ [373], K. Dilcher, et al. [144]. (For an electronic, updated version, see K. Dilcher [142]). The Bernoulli and Stirling numbers are important in some inversion formulas. Forexample, if n n F(n) = f (k) − λf (n)(n = 0, 1,...) k=0 k then + 1 n 1 n + 1 f (n) = F(k)B + − , λ = , + n 1 k if 1 n 1 k=0 k and − n n nm k!S(n − m, k) f (n) =− F(m) , if λ = 1; (λ − )k+1 m=0 m k=0 1 and reciprocally, see Z. Sun [431]. Let s∗(n, k) denote the number of permutations of an n-element set that have k disjoint cycles. Then

~~s∗(n, n) = s∗(n, 1) = 1, s∗(n, k) = 0 for n < k, and~~

s∗(n, k) = s∗(n − 1, k − 1) + (n − 1)s∗(n − 1, k)(16) The numbers s∗(n, k) are called the unsigned Stirling numbers of the ﬁrst kind. These numbers satisfy a counterpart of (2), namely:

n ∗ k xn = s (n, k)x ,(17) k=0 ∗ ∗ where xn is the rising factorial power, and s (n, 0) = 1ifn ≥ 1, s (0, 0) = 0. The numbers s∗(n, k) are nonenegative in contrast with the (normal) Stirling numbers of the ﬁrst kind s(n, k) given by

~~s∗(n, k) =|s(n, k)|=(−1)k+ns(n, k)(18)~~

~~Then in place of (17) one can write~~

~~n k (x)n = s(n, k)x (19) k=0 One has the recurrence formulae:~~

~~s(n, k) = s(n − 1, k − 1) − (n − 1)s(n − 1, k)(n, k ≥ 1)(20)~~

~~462 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS with s(n, 0) = s(0, k) = 0, s(0, 0) = 1 = s(n, n);~~

~~− n 1 n! ks(n, k) = (− )n−l−1 s(l, k − ), ( ) 1 ( − ) 1 21 l=k−1 l! n l !~~

n s(n + 1, k + 1) = (−1)n−l (l + 1)(l + 2)...(n)s(l, k), (22) l=k n l (n − k)s(n, k) = (− )l−k s(n, l), ( ) 1 − 23 l=k+1 k 1 n s(n, k) = s(n + 1, l + 1)nl−k (24) l=k One can write also tn tn s∗(n, k) xk = (1 − t)−x = x ,(25) n,k≥0 n! n≥0 n!

~~n n t k x t s(n, k) x = (1 + t) = (x)n ,(26) n,k≥0 n! n≥0 n!~~

~~n s∗(n, k)un−k = (1 + u)(1 + 2u)...(1 + (n − 1)u), (27) k=1 n s(n, k)un−k = (1 − u)(1 − 2u)...(1 − (n − 1)u)(28) k=1 From (26) one can deduce~~

~~tn logk(1 + t) s(n, k) = ,(29) n≥k n! t! which is similar to (3). From (27) one obtains~~

~~n s∗(n + 1, k + 1)xk = (x + 1)(x + 2)...(x + n), (30) k=0 n s∗(n + 1, n + 1 − l)ul = (1 + u)(1 + 2u)...(1 + nu). (31) l=0~~

~~463 CHAPTER 5~~

~~The numbers s(n, k) may be expressed in terms of the Stirling numbers of the second kind (due to O. Schlomilch¨ [394], see also [237]):~~

~~− n k n − 1 + m 2n − k s(n, k) = (− )m S(n − k + m, m)() 1 − + − − 32 m=0 n k m n k m~~

~~Let L∗(n, k) denote the number of ways to partition an n-element set into k nonempty (linearly ordered) queues. Then L∗(n, n) = L∗(n, 1) = 1, L∗(n, k) = 0for n < k,and~~

~~L∗(n, k) = L∗(n − 1, k − 1) + (n + k − 1)L∗(n − 1, k)(33)~~

~~These numbers are called the unsigned Lah numbers (or Stirling numbers of the third kind, see [457]). In 1954 I. Lah [269] proved that~~

~~n ∗ xn = L (n, k)(x)k (34) k=0 where L∗(n, 0) = 1ifn = 0, L∗(0, 0) = 0. The (ordinary) Lah numbers L(n, k) are deﬁned by~~

~~L(n, k) = (−1)n L∗(n, k)(35)~~

~~Then one has~~

n n k k xn = (−1) L(n, k)(x)k; (x)n = (−1) L(n, k)xk; k=0 k=0 see also I. Lah [270] and M. Petkovsekˇ and T. Pisanski [347] (where the history of Lah numbers is also considered). For connections to Physics, see M. Schork [396].

2 Identities for Stirling numbers We mention several other identities for the Stirling numbers of both kinds: n+1 0, n = m s(n, i)S(i, m) = (36) 1, n = m i=m n+1 0, n = m S(n, i)s(i, m) = (37) 1, n = m i=m

~~464 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

((36) and (37) are called also the orthogonality properties of the Stirling numbers of both kind); + n 1 i s(n, i) = s(n − 1, m) + s(n − 1, m − 1), (38) i=m m and by inversion n m s(n, k) = (− )m+k+1 s(n − , m)() 1 − 1 39 m=k−1 k 1

~~These lead also to other formulae (see [237], pp. 187-189):~~

~~+ n 1 n s(i + 1, m + 1)S(n, i) = (−1)m+n ,(40) i=m m~~

+ n 1 n s(i, m)S(n + 1, i + 1) = ,(41) i=m m + n 1 n S(i, m) = S(n + 1, m + 1), (42) i=m i + n 1 n is(i, m)S(n, i) = (− )n+m ,() 1 − 43 i=m m 1 + n 1 n (− )n+i S(i, m) = mS(n, m), ( ) 1 − 44 i=m i 1 + n 1 i (− )m+i s(n, i) = n · s(n, m)() 1 − 45 i=m m 1 n (n − 1)! s(n, i + 1)S(i, m) = (−1)n+m+1 (46) i=m m! For the unsigned Stirling numbers s∗(n, k) of the ﬁrst kind, one has (see Ch. Jordan [237], pp. 153, 165, 194, 339): n ∗( , ) ∗( , + ) s i m = s n m 1 ,() ( − ) 47 i=m i! n 1 ! ∞ ∗( , ) s n k = ,() ( + ) 1 48 k=0 k 1 !

~~465 CHAPTER 5~~

∞ s∗(n, k) = ζ(n + )( ζ ), ( ) · 1 where is the Riemann zeta function 49 k=1 k k! ∞ 1 s∗(n, k) · = ζ(n + ) − .() ( + ) 1 1 50 k=1 k k 1 ! By using the metric theory of Pierce expansions, J. O. Shallit [398] has proved that

~~∞ − s∗(n, k) m1 s∗(n, k) 1 n = − = s∗(i, m) n ≥ , m ≥ ; ( + ) 1 ( + ) for 1 1 k=m k 1 ! k=0 k 1 ! m! i=1~~

∞ s∗(n, k) H(k) = ζ( ) + ζ( ) + ···+ζ(n + ), ( ) ( + ) 2 3 1 51 k=1 k 1 ! ∞ s∗(n, k) H(k + ) = n + ,() 1 ( + ) 1 52 k=1 k 1 ! 1 1 where H(k) = 1 + + ···+ is the kth harmonic number. 2 k As an application of (51), a formula for ζ(3), due to W. E. Briggs et al. [61] can be obtained: ∞ 1 H(k) ζ(3) = (53) 2 2 k=1 k Another consequence of (51) is

~~∞ ( )( ( − ) − ) ζ( ) = H k H k 1 1 ( ) 3 ( + ) 54 k=1 k k 1~~

For harmonic sums and the zeta function, see also [181]. For the arithmetic properties of H(n),see e.g. D. W. Boyd [60], A. Gertsch [182]. For the computation of Euler’s constant via H(n),see E. A. Karatsuba [243]. See also J. Sandor´ [383]. By using certain basic Stirling polynomials, H. Niederhausen [327] obtains

i + j + qn s(i + j + ln, i + j + (l − 1)n) = i + j + (l − 1)n n + + + + ( − ) −1 + = i j ln i j l 1 n i qk · j · + + ( − ) + ( − ) + ( − )( − ) k=0 i lk i l 1 k i l 1 k j l 1 n k ·s(i + lk, i + (l − 1)k)s( j + l(n − k), j + (l − 1)(n − k)) (55)

466 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS where i, j, q, l, n are nonnegative integers so that all expressions in (55) are well deﬁned. For l = 1, q = 0andm = i + j + n,(55) implies the well known result:

~~− i + j mj m s(m, i + j) = s(k, i)s(m − k, j), (56) i k=i k see G. C. Rota et al. [371]. For l = q = 0, m = i + j − n, the identity~~

~~− − i + j 1 i + j i m 1 i j s(i + j, m) = s(i, k) s( j, m − k)() − 57 i m k=m− j k k m k follows, where i < m, j < m, i + j ≥ m. By applying Nielsen polynomials, one obtains~~

− m − i − j i + j mj m k − i s(m, i + j) = s(k, i)s(m − k, j), ( ) + 58 i j i k=i+1 k i for i + j ≤ m, which is analogues to (56). There can be obtained identities for S(n, k), too; as well as mixed identities. We quote only the following:

i + j − qn S(i + j − (l − 1)n, i + j − ln) = i + j − ln n + − ( − ) + − −1 − = i j l 1 n i j ln · i qk · j · − ( − ) − − − ( − ) k=0 i l 1 k i lk i lk j l n k ·S(i − (l − 1)k, i − lk)S( j − (l − 1)(n − k), j − l(n − k)) (59) For l = q = 0, m = i + j − n one obtains an identity dual to (56), while for l = 1, q = 0, m = i + j + n we get an identity dual to (57). Forsimilar identities, see H. W. Gould [196], too. Identities involving the Bernoulli and Stirling numbers of the ﬁrst kind have been proved by S. Shirai and K. Sato [399]. Let m ≥ 0, 1 ≤ k ≤ m + 1beintegers. Then one has B k s(m, k − 1) = k + m1 s(p + 1, q + 1)...s(p + 1, q + 1) = m (− )t 1 1 t t ( ) ! 1 ( + ) ...( + ) 60 +···+ = + +···+ = p1 1 ! pt 1 ! t=1 p1 pt m 1 q1 qt k ≥ ( ≥ ) pi 1 qi 0

~~467 CHAPTER 5~~

~~Let f (x, a) = a/(1 − (1 − x)a) for a = 0. By comparing the coefﬁcients of two distinct expansions of f (x, a) about x = 0, Shirai and Sato prove also that~~

~~+ + m1 B m1 1 k s(m, k − ) = (− )nm (− )t ( ) 1 1 ! 1 ( + )...( + ) 61 k +···+ = + p1 1 pt 1 k=1 t=1 p1 pt m 1 ( ≥ ) pi 1~~

~~(k) In [338] the Bernoulli numbers of order k, Bn are deﬁned by k k ( − )( − )...( − ) = (k+1) r ( ) x 1 x 2 x k Bk−r x 62 r=0 r~~

~~On the other hand, by (19),~~

~~k (x − 1)(x − 2)...(x − k) = s(k + 1, r + 1)xr , r=0 which by (62) implies k − 1 s(k, k − n) = B(k) (k ≥ n + 1)(63) n n~~

~~The Bernoulli numbers of higher order satisfy the recurrence n−1 1 n + 1 ( ) (−1)n B(k) = (−1)i (k − i) B k (64) n ( + ) i n n 1 i=0 i~~

(k) Forcertain special values of Bn ,see [311]. In 1976 L. Carlitz [77] introduced the numbers b(n, k) and B(n, k) by k n + j − 1 s(n, n − k) = b(k, j) , j=1 2k (65) k n + j − 1 S(n, n − k) = B(k, j) j=1 2k Then one has

~~b(k, j) = (2k − j)b(k − 1, j − 1) + jb(k − 1, j) for 1 ≤ j ≤ k, (66) b(1, 1) = 1, b(1, j) = 0 for j > 1;~~

468 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS and B(k, j) = (k − j + 1)B(k − 1, j − 1) + (k + j − 1)B(k − 1, j) for 1 ≤ j ≤ k, B(1, 1) = 1, B(1, j) = 0for j > 1. (67) In fact, b(k, j) = B(k, k − j + 1) for 1 ≤ j ≤ k.(68) The following explicit formulae are true: j 2k + 1 B(k, k − j + ) = (− ) j−t S(t + k, t); 1 1 − t=0 j t (69) j 2k + 1 b(k, k − j + ) = (− ) j−t s(t + k, t) 1 1 − t=0 j t G. Mastroianni [296] used the Stirling numbers in order to study certain linear operators from Approximation theory. He then proved certain identities for generalized Stirling numbers. J. L. Littlejohn and R. Wellman [288] give new applications of S(n, k) to operatorial theory and differential equations. For identities concerning determinants of matrices whose entries are s∗(n, k) and S(n, k), see B. S. Chandramouli [101]. Various identities are proved for the generalized Stirling numbers (see the next section).

3 Generalized Stirling numbers We now study certain generalized Stirling numbers. A. A. Broder [63] deﬁned regular Stirling numbers (or r-Stirling numbers for short) Sr (n, k) for 0 ≤ r ≤ k ≤ n,asthe number of partitions of the set {1, 2,...,n} into k nonempty disjoint subsets, such that the numbers 1, 2,...,r are in distinct subsets. It is easy to see that S0(n, k) = S1(n, k) = S(n, k) = Stirling number of the second kind. Broder proved that

R(n, k, r) = Sr (n + r, k + r), (70) n n where R(n, k, x) = S(m, k)xn−m (x ∈ R) are the Stirling polynomials of m=0 m the second kind introduced by L. Carlitz [85]. R(n, k, x) has an expression by means of divided differences (see [63], [85]) R(n, k, x) = [x, x + 1,...,x + k]tn,(71)

469 CHAPTER 5 implying n Sr (n + r, k + r) = [r, r + 1,...,r + k]t (72) Fordivided differences and combinatorial identities, see L. Verde-Star [464]. The recurrence formula R(n, k, x) = (x + k)R(n − 1, k, x) + R(n − 1, k − 1, x)(0 < k ≤ n)(73) wasgiven by Carlitz [85]. More relations were given by E. Newman [320]: R(n, k, x + 1) = (k + 1)R(n, k + 1, x) + R(n, k, x)(0 ≤ k ≤ n); (74) − − nk 1 n (k − )kR(n, k, x) = n(k − )R(n − , k − , x) + (x + k − )· 1 1 1 1 + [ 1 j=0 j k ·( j + k)R( j + k − 1, k − 1, x) − ( j + 1)R( j + k, k − 1, x)] (1 ≤ k ≤ n)(75) Neuman proves also inequalities for R(n, k, x).For example: (x + k)n − k(x + k − 1)n (x + k)n ≤ R(n, k, x) ≤ , for 0 ≤ k ≤ n,(76) k! k! with equality only for k = 0; R(n + 1, k, x)R(n − 1, k, x) ≤ (R(n, k, x))2 ≤ n n + 1 − k ≤ · R(n + 1, k, x)R(n − 1, k, x), (77) n − k n + 1 where 0 ≤ k < n. The ﬁrst inequality tells us that the sequence (R(n, k, x))n is logarithmically concave. If 1 ≤ k ≤ n, then kR(n, k, x) ≥ nR(n − 1, k − 1, x); (78) and if 1 < k < n, then n − k xR(n − 1, k, x)< R(n, k, x)<(x + k)R(n − 1, k, x)

~~1 1 R(k + 1, k, x) ≥ R(k + 2, k, x) 2 ≥ R(k + 3, k, x) 3 ≥ ... (79)~~

This give inequalities for the r-Stirling numbers Sr (n, k).For example, (76) implies (r + k)n − k(r + k − 1)n (r + k)n ≤ S (n + r, k + r) ≤ for k ≤ n,(80) k! r n! which for r = 0reduces to a result by H. Wegner [480].

~~470 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

By deﬁning the rth complete symmetric functions hr (r ≥ 0) by = ( ,..., ) = i0 ... in ( ) hr hr x0 xn x0 xn 81 i0+···+in=r n + r (where i ,...,i ∈{0, 1,...,r} and the sum involves terms and 0 n r n+1 (x0,...,xn) ∈ R+ ), in [322] E. Neuman obtains the identity

~~Sr (n + r, k + r) = hn−k(r, r + 1,...,r + k)(82)~~

Here certain new identities and inequalities for Sr ,aswell as for the q-binomial coefﬁcient of Gauss are deduced. The generalized nth symmetric mean is n + r H (x ,...,x ) = xi0 ...xin (83) r 0 n n 0 n i0+···+on=r

~~K. V. Menon [303] proved the logarithmic convexity of the means (83) for r = 1, 2, 3. In 1979 D. W. DeTempe and J. M. Robertson [443] have proved that~~

~~2( , ) ≤ ( , ) ( , )() Hr x0 x1 Hr−1 x0 x1 Hr+1 x0 x1 84~~

~~The logarithmic convexity of Hr (x0,...,xn) without any restriction to r and n has been proved by E. Neuman in [323]. (85) He prove also the strong logarithmic concavity of (S(n, k))n, i.e.~~

~~S2(n, k)>S(n − 1, k)S(n + 1, k)(2 ≤ k ≤ n)(86)~~

~~We note that the strong logarithmic concavity of the sequence (S(n, k))k has been established by L. H. Harper [213] and E. H. Lieb [287]:~~

~~S2(n, k)>S(n, k − 1)S(n, k + 1)(2 ≤ k ≤ n)(87)~~

In the proofs of (85) and (86) the B-splines by H. B. Curry and I. J. Schoenberg [125] are applied. ( , ) r An identity connecting Sr n k to the ”hyperharmonic numbers” Hn is r = ( + , + )/ ( ) Hn Sr n r n 1 n! 82

~~1 n 0 = r = r−1 , ≥ where Hn , Hn Hi for r n 1. n i=1~~

~~471 CHAPTER 5~~

For (82 ) and other identities, e.g. m m n! S (n + r, k) = S − (n + r − m, k − t) r ( + − ) r t t=0 t n t m ! for 0 ≤ m ≤ r,see A. T. Benjamin et al. [42]. r The numbers Hn have been introduced by H. J. Conway and R. K. Guy [118], and in fact, + − r n r 1 H = (H + − − H − ), n r − 1 n r 1 r 1 where Hn are the ordinary harmonic numbers. For another class of Stirling numbers, namely p-Stirling numbers,see R. Merris [306]. B. The Stirling numbers of fractional order have been introduced and studied by P. L. Butzer, M. Hauss and M. Schmidt [72]. Let be the Euler gamma function, and define for a ∈ R the binomial coefficient functions x (x + 1) = (x ∈ R, x ∈ Z−)(88) a (a + 1)(x − a + 1) where Z− ={−1, −2, −3,...}.The falling and rising factorial functions will be (x + 1) (x + a) (x) = (x ∈ Z−), x = (x + a ∈ Z−)(89) a (x − a + 1) a (x) 0 Z− = Z− ∪{ } where 0 0 . The Stirling functions s(a, k)(k = 0, 1,...)of first kind will be defined via, ∞ k (x)a = s(a, k)x (|x| < 1)(90) k=0 or equivalently by 1 d k s(a, k) = (x)a (a ∈ R, k = 0, 1,...) (91) k! dx x=0 We note that the series in (90) is well-defined for all a ∈ R on account of the properties of the gamma function. The series is absolutely and uniformly convergent for |x| < 1. For all a ∈ R, k ≥ 0 there hold the relations (see [72]) ( , ) ∞ ( , ) π( − ) s a k = s j k · sin a j ,() ( + ) π( − ) 92 a 1 j=k j! a j

472 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS and ∞ a s(a, k) = s( j, k)s(a − j, 0)(a ∈ Z−)(93) j=k j both series being uniformly convergent in a. The Stirling functions of the ﬁrst kind s(a, k) satisfy the triangular recurrence relation s(a, k) = s(a − 1, k − 1) − (a − 1)s(a − 1, k), 1 (94) s(a, 0) = (1 − a)s(a − 1, 0) = (1 − a) Further, for a > n ≥ 0 they satisfy

~~n j n+1 s(a + 1, k) = (−1) (a) j s(a − j, k − 1) + (−1) (a)n+1s(a − n, k)(95) j=0~~

There holds the complex integral representation 1 (w + 1) 1 s(a, k) = · dw, (96) k+1 2πi K (w − a + 1) w where K is a Jordan-curve in C for which 0 ∈ intK, and K ⊂{z ∈ C : |z| < 1}. The Stirling functions of the second kind S(a, k) will be deﬁned in analogy with (91) by 1 S(a, k) = k xa (a ≥ 0, k = 0, 1,...) (97) k! x=0 where is the forward difference operator, namely f (x) = f (x + 1) − f (x), j f (x) = ( j−1 f )(x), j ∈ N. The following formulae hold true: 1 k k S(a, k) = (−1)k+ j j a,(a ≥ 0, k ≥ 0), (98) k! j=0 j 1 (w − 1)a S(a, k) = dw(a > 0), (99) 2πi C (w − 1)...(w− k − 1) where C is an arbitrary rectiﬁable Jordan-curve in the semi-plane Re z > 1 containing all n ∈{2, 3,...,k + 1}. For a > 0 there holds ∞ a z = S(a, k)(z)k,(100) k=0

~~473 CHAPTER 5~~

1 the abscissa of convergence being λ ≤ max 0, a − . The series converges ab- 2 solutely for |z|≥1 + λ and uniformly on all compact subsets of the half plane Re z ≥ λ + ε for arbitrary ε>0. For a > 1, k ∈ N they satisfy the triangular recurrence relation

~~S(a, k) = kS(a − 1, k) + S(a − 1, k − 1), (101) S(a, 0) = S(0, k) = 0, S(0, 0) = 1.~~

~~We note also the generating functions~~

~~∞ n k x 1 b S(a + bn, k) = (−1)k+ j j ae j x (a, b ≥ 0) n=0 n! k! j=0 (102) ∞ ∞ ka S(a, k)xk = e−x xk (a ≥ 0, x ∈ R) k=0 k=0 k!~~

~~One has also S(a, k) 1 S(a + 1, k) lim = , lim = k.(103) a→∞ ka k! a→∞ S(a, k)~~

In [70] Butzer and Hauss prove several results for S(α, k), α ≥ 0, k ∈ N, viewed as a function of α. Among the results obtained are proofs of the continuity and dif- ferentiability of S, real integral representations, a representation in terms of the Weyl derivative of fractional order α,and connections with Bernoulli, Stirling and Bern- stein polynomials, and with Bernoulli numbers of fractional order. C. Starting from a given sequence t = (tk)k≥0,in1972 L. Comtet [117] has given the following generalization of classical Stirling numbers. The Stirling numbers of the ﬁrst kind st (n, k) associated to the sequence t are deﬁned by

n−1 n k (x − tk) = st (n, k)x ,(104) k=0 k=0 and the Stirling numbers of the second kind, St (n, k), associated to the sequence t are given by n n x = St (n, k)(x − t0)...(x − tk−1)(105) k=0

~~Particularly, for tk = 1 one gets the binomial coefﬁcients, while for tk = k the usual Stirling numbers.~~

~~474 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

I. Aldanondo [17] has given a generalization of ﬁnite differences by 0 = , n+1 = n − n Dt xm xm Dt xm Dt xm+1 tn Dt xm where t = (tk)k≥0 is a given sequence. Deﬁning the shift, resp. identity operators E and I by Exm = xm+1, Ixm = xm, one can write:

n−1 n = ( − ) Dt xm E tk I xm k=0 Therefore n n n = ( , ) , = ( , ) k ( ) Dt xm st n k xm+k xm+n St n k Dt xm 106 k=0 k=0 S. Toader and Gh. Toader [446] have introduced the Stirling numbers associated = ( ) 0 = n+1 = to a matrix T tn,m n,m≥0.Define the finite differences DT xm xm, DT xm n − n ( , ≥ ) ( , , ) ( , , ) DT xm+1 tn,m DT xm n m 0 ,and define sT n m k and ST n m k by n n n = ( , , ) , = ( , , ) k ( ) DT xm sT n m k xm+k xm+n ST n m k DT xm 107 k=0 k=0 n From the definition of DT and by (107) one can deduce that n sT (n, m, n) = 1, sT (n, m, n − 1) =− tn−k,m+k−1,..., k=1 n sT (n, m, 0) = (−1) tn−1,m ...t0,m, and the triangular recurrences

~~sT (n + 1, m, k) = sT (n, m + 1, k − 1) − tn,msT (n, m, k), (108) where sT (n, m, k) = 0fork < 0ork > n. Similarly, for the numbers ST (n, m, k) we have~~

ST (n + 1, m, k) = ST (n, m + 1, k − 1) + tk,m ST (n, m + 1, k)(109) The following vertical recurrences are valid: n n n− j sT (n + 1, m, k) = (−1) sT ( j, m + 1, k − 1) ti,m j=k−1 i= j+1 (110) n m+n− j−1 ST (n + 1, m, k) = ST ( j, m + n − j + 1, k − 1) tk,i j=k−1 i=m

~~475 CHAPTER 5~~

~~If one considers the matrices~~

sT (m) = (sT (n, m, k))n,k≥0, sT (m) = (ST (n, m, k))n,k≥0, then sT (m)ST (m) = S)T (n)sT (n) = I (111) These are generalizations of the classical results. For generalizations of orthogonality properties like (111), see K. S. Miller [307], H. W. Gould [197], S. Tauber [437] (”quasi-orthogonality”), [438] (”bilinearly recurrent orthogonality”). For Stirling numbers generalization via methods of generating functions, and recurrent sequences, see e.g. L. Toscano [449], J. M. Sixdeniers, K. A. Penson, and A. I. Solomon [403]. The matrix approach is adopted also by T. Bickel [48], where one can ﬁnd symmetries and dualities e.g. between Lah and Stirling numbers, between matrices constant on rows and those constant on columns, between matrices and their inverses; etc. D. In 1979 L. Carlitz [86], and in 1985 F. T. Howard [221] have introduced and studied the degenerate weighted Stirling numbers. Let 0 ≤ k ≤ n be integers, λ, θ = 0real numbers, |x| < 1. Then deﬁne the numbers s(n, k,λ|θ) and S(n, k,λ|θ) by

∞ 1 − (1 − x)θ k xn ( − x)θ−λ = k s(n, k,λ|θ) ( ) 1 θ ! 112 n=k n! and ∞ xn (1 + θx)λ/θ ((1 + θx)1/θ − 1)k = k! S(n, k,λ/θ) (113) n=k n! There are the Howard degenerate weighted Stirling numbers [221], which could be deﬁned also by basis-transformations. E.g. by

~~n (x + λ|θ)n = S(n, k,λ/θ)(x)k,(114) k=0~~

~~n−1 where (t|θ)n = (t − kθ) (n ≥ 1). k=0 In fact the equivalence between (113) and (114) may be verifed by means of the recurrence relations~~

~~476 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

The Carlitz degenerate Stirling numbers [86] s(n, k|θ) and S(n, k|θ) are special cases when λ = 0. For λ = 0, θ → 0 one reobtains the classical Stirling numbers s(n, k) and S(n, k). See also [87]. Another related class of generalized Stirling numbers, called Dickson-Stirling numbers have been deﬁned by (see L. C. Hsu, G. L. Mullen and P. J. S. Shiue [226])

~~n Dn(x,α)= S(n, k,α)(x − α)k (n = 1, 2,...) (116) k=0 where~~

~~/ [n 2] n n − i D (x,α)= (−α)i xn−2i , D (x,α)= . n − with 0 2 i=0 n i i~~

In [227] the numbers S(n, k,λ|θ)and S(n, k,α)have been applied in order to introduce two kinds of generalized Eulerian polynomials and numbers, which are associated with the construction of certain summation formulas for extended arithmetic- geometric series. For the numbers S(n, k,α)we quote (see [227]):

∞ n S(n, j,α)x j D (k,α)xk = xα j! ,(117) n ( − ) j+1 k=α j=0 1 x where α ≥ 0isagiven integer and |x| < 1. Another class of degenerate Stirling numbers, introduced via certain exponential Bell polynomials appears also in [222]. Hsu and Shiue [228] have deﬁned a class of generalized Stirling numbers S(n, k; α, β, γ ) as follows:

n (x|α)n = S(n, k; α, β, γ )(x − γ |β)n (118) k=0 where (x,α)n is deﬁned as in (114). For some combinatorial and statistical applications of these numbers, see R. B. Corcino, L. C. Hsu and E. L. Tan [120]. For combinatorial applications of some weighted Stirling and related numbers, see also C. A. Charalambides [102]. In [229] L. C. Hsu and H. Q. Yu deﬁned a ”Stirling- type pair” {S1(n, k), S2(n, k)}={S(n, k; α, β, r), S(n, k; β,α,−r)}.Forα = 1 the S1 and S2 are essentially Howard’s degenerated weighted Stirling numbers. In [229] quasi-Schlomilch¨ formulas, while in [120] orthogonality relations, generating functions, congruence properties, etc., are worked out.

~~477 CHAPTER 5~~

~~E. Acommon generalization of the Stirling numbers, and Lah’s numbers are the Sα,β (r, m) numbers studied by E. G. Tsylova [457]. These are deﬁned by the identity~~

~~m m−1 m−1 Sα,β (r, m) (x + βh) = (x + αk)(119) r=0 h=0 k=0~~

It is easy to see that from S−1,0(k, m), S0,−1(k, m) and S1,−1(k, m) one can reobtain the numbers s(m, k), S(n, k) and L(n, k), respectively. Starting with the generating function ∞ 1 1 − (1 − αx)β/α r xm = Sα,β (r, m) ( ) β 120 r! m=r m! and applying the Cauchy residue theorem, Tsylova proves also that if α = β, and r, m →∞so that 0 < m − r = O(m1/2), then − m 1 m r Sα,β (r, m) = (α − β)r · (1 + o(1)) (121) r 2 For generalized Stirling and Lah numbers, see also C. G. Wagner [470]. F. General factorial numbers were introduced by L. Verde-Star [465], or (n) H. Niederhausen [328]. The general factorial powers xa,b and factorial numbers f a,b(n, k), Fa,b(n, k) (where n, k are nonnegative integers, and a = b real numbers) are deﬁned by ( ) x x − nb x n = − 1 a,b − − a b a b n−1 and (n) = (a,b)( , ) k ( ) xa,b f n k x 122 ≥ k 0 n = (a,b)( , ) (k) ( ) x F n k xa,b 122 k≥0 appear in [328]. Here (·)n−1 denotes a falling factorial. Since f 1,0(n, k) = s(n, k) and F 1,0(n, k) = S(n, k), the factorial numbers f a,b(n, k) and Fa,b(n, k) are generalizations of the Stirling numbers. Some properties of factorial numbers are a,b a,b a,b a,b f (n, 0) = F (n, 0) = δn,0 and F (n, k) f (k, m) = δn,m (123) k≥0

(i.e. orthogonality), where δn,m is the Kronecker delta; f a,b(n, n) = (a − b)−n; Fa,b(n, n) = (a − n)n; ck f ac,bc(n, k) = f a,b(n, k); cn Fac,bc(n, k) = Fa,b(n, k); (124) f a,b(n, k) = (−1)n f b,a(n, k); Fa,b(n, k) = (−1)k F b,a(n, k).

~~478 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

The numbers f a,b(n, k) can be expressed in terms of Stirling numbers of the ﬁrst kind: n j − 1 f a,b(n, k) = s(n − , j − ) (b( − n) − a) j−k(a − b)− j , 1 1 − 1 j=k k 1 ( ) 125 n j − 1 f a,b(n, k) = s(n, j) (−bn) j−k(a − n)− j , − j=k k 1 for all n ≥ k ≥ 1. Closed form representations of Fa,b(n, k) and f a,b(n, k) do exist, butare slightly complicated for the latter (see [329])

k ( ja + (k − j)b)n Fa,b(n, k) = (− )k− j ,() 1 ( − ) 126 j=0 j! k j ! n + k k 2k 1 f a,b(n, k) = (n − k)(a − b)k−n · − − ( + ) n k j=0 k j j! n j + j j b j k · (− )γ γ + j γ 1 − γ =0 a b 1 for n ≥ k ≥ 0. For a = =−b one obtains the so called central factorial numbers, 2 studied by P. L. Butzer, et al. [73], where (126) appears in this particular case. But the non-central Stirling numbers by M. Koutras [260] are covered, too. Indeed, (a − b)−n Fa,b(n, k) is a non-central Stirling number, with non-centrality parameter k a,b kb/(b − a),inthe sense of [260]. Similarly, (a − b) f (n, k) = sc(n, k) + (n − 1 − c)sc(n − 1, k), where sc(n, k) is the non-central Stirling number of the ﬁrst kind with nb non-centrality parameter c = − 1. b − a For f 1+c,c(n, m) with n ≥ m ≥ 0 there exist identities, as: − n ki (−1)n mn! ∗ 1 i(1 + c) f 1+c,c(n, m) = ,(127) ( + )m ki 1 c i=1 i ki ! i where the sum is taken for k1 + ···+kn = m,1· k1 + ···+n · kn = n and ki ≥ 0. For c = 0 this is a well-known statistic on permutations, where ki is the number of cycles of length i. Alternatively,

~~m 1+c,c n−m n! f (n, m) = (−1) (l (1 + c) − 1)( − )/l ! (128) m! i li 1 i l1+···+lm =n i=1~~

~~479 CHAPTER 5~~

where all li ≥ 1. Finally, there is a third sum of products: − − (n − 1)! n 1 n(c + 1) − l − 1 m1 1 f 1+c,c(n, m) = (−1)n−m (m − 1)! n − 1 − l li l=m−1 l1+···+lm−1=l i=1 (129) M. Kontras [260] derived the above sum for noncentral Stirling numbers of the first kind. For proofs of (127)-(129), see H. Niederhausen [330]. For Fa,b(n, k) there exists the generating function formula (a, b, d ∈ R): tn Fa,b(n, k)xk = exp{x(eat − ebt )},(130) n≥0 k≥0 n! implying that tn 1 Fa,b(n, k) = (eat − bat)k,(131) n≥k n! k! − i + j n j n Fa,b(n, i + j) = Fa,b(m, i)Fa,b(n − m, j), (132) i m=i m n n Fa,b(n, k) = (dk)n− j Fa−d,b−d ( j, k), (133) j=k j n Fa,b(n, k) = Fa,d (i, j)F d,b(n − i, k − j)(134) i, j i Here formula (133) is equivalent to an identity of Verde-Star [465]. For (130)- (132), see [328], and for (133)-(134), see [330]. The motivation for defining generalized factorial powers in the form from (122) comes from the recurrence ( + )(n) − ( + )(n) = (n−1) x a a,b x b a,b nxa,b ≥ ( )(n) = δ for all n 1. Together with the initial values 0 a,b 0,n, this recurrence shows {( )(n) ≥ } that x a,b : n 0 is a sequence of binomial type. For combinatorial and statistical applications of f a,b(n, k) and Fa,b(n, k),see [328], [331]. In [331] it is described a new bijection from n-permutations with η-colored head and τ-colored tail records to positive integer n-sequences where each member stays between certain bounds. This bijection extends a construction given by R. P. Stanley [416] for Stirling numbers. F. The theory of q-analogs has a long history, going back to Gauss who studied the q-binomial coefficients n n (qn − 1)(qn−1 − 1)...(qn−k+1 − 1) = = , for q > 1; k q k (q − 1)(q2 − 1)...(qk − 1)

~~480 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

n = for q = 1, where q is a positive integer. Recently there has been considerable k interest in q-Stirling numbers. The q-Stirling numbers of the first kind, s[n, k], were introduced in H. W. Gould’s paper [198]. He studied also the q-Stirling numbers of the second kind S[n, k], but these numbers were earlier introduced by L. Carlitz [90], [91]. The numbers S[n, k]were studied also by S. C. Milne [308]. I. M. Gessel [185] gave an interpretation for s[n, k]asthe generating function for an inversion statistic. For the numbers S[n, k], S. C. Milne [309] showed that these numbers could be viewed in terms of inversions on partitions, and B. E. Sagan [375] showed that there was also aversion of the major index. S. C. Milne [309] and M. Wachs and D. White ([376]) proved that the q-Stirling numbers S[n, k]count restricted growth functions using various statistics. Milne gave also a vector space interpretation as follows. Let q be a prime power, and let V be a vector space over the Galois field Fq .Givenany set of vectors S ⊂ V ,letS denote the linear span of S.Now any sequences of one dimensional subspaces of V (i.e. lines) l1, l2,...,ln determines a flag V0 ⊂ V1 ⊂ ···⊂ Vk, obtained by deleting the repeated subspaces in 0⊂l1⊂···⊂l1, l2,...,ln. Now fix the above flag of dimension k, and let {v1,...,vk} be a basis for Vk such that {v1,...,vi } is a basis for Vi for all i.Then the group of upper unitriangular matrices in this basis acts on the sequences of n lines generating the given flag of dimension k. Milne [309] showed that S[n, k]isthe number of orbits under this action, regardless the initial choice of a flag. The q-analogue of a positive integer n is defined by [n] = 1 + q +···+qn−1 The numbers s[n, k] can be defined by the recurrence s[n, k] = s[n − 1, k − 1] − [n − 1]s[n − 1, k] for n ≥ 1 (135) and s[0, k] = δ0,k. The signless q-Stirling numbers of the first kind s∗[n, k]satisfy ∗ ∗ ∗ s [n, k] = s [n − 1, k − 1] + [n − 1]s [n − 1, k]forn ≥ 1 ∗ (135 ) s [0, k] = δ0,k Finally, the numbers S[n, k]aredefined by S[n, k] = S[n − 1, k − 1] + [k]S[n − 1, k] for n ≥ 1 (136) S[0, k] = δ0,k n For combinatorial interpretations involving inversions of , s∗[n, k], S[n, k], k see B. E. Sagan [376]. For partition lattice q-analogs related to q-Stirling numbers,

481 CHAPTER 5 see T. A. Dowling [146] and C. Bennett, K. J. Dempsey and B. E. Sagan [43]. For certain other statistics on the set of partitions of {1, 2,...,n} (e.g. the mak or mak’ functions of Steingrimsson) involving S[n, k], see G. Ksavrelof and J. Zeng [263]. There exist many identities involving the q-Stirling numbers. For example (see [198]) k k + n j + n S n, k = (q − )−k (− )k− j , [ ] 1 1 − j=0 k j j ( ) 137 k n − j n s n, k = (q − )−k (− )k− j q j( j+1)/2 [ ] 1 1 − j=0 k j j

~~The following two identities are due to Carlitz [90], [91]: n n n = (q − 1) j−k S[ j, k], k j=k j (137 ) n m [m]n = qk(k−1)/2 [k]!S[n, k] k=0 k~~

Applying various Mobius¨ inversions to the ﬁrst formula of (137 ),wecan deduce other equivalent identities: n (n−l) n j n− j ∗ q 2 = (q − 1) s [n, j], l j=l l n − − n j (q − 1)n l S[n, l] = (−1)n j , (137 ) j=l j l n − − ∗ j−l j (n j) n (q − 1)n l s [n, l] = (−1) q 2 j=l l j

For combinatorial proofs of these identities (including (137 )), see A. de Medicis´ and P. Leroux [302]. The q-analogue of a Frobenius identity is due to A. M. Garsia and J. B. Remmel [175]: n , k n m (k) S[n k][k]!x [m] x = q 2 ( − )( − )...( − k) m≥0 k=0 1 x 1 xq q xq

~~482 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~The double generating functions are due to I. Gessel [185] and to Garsia and Remmel [176], respectively:~~

~~∞ xn 1 − qk x s∗[n, k]uk = ; − ( + ( − ) ) k n≥0 k≥0 [n]! k=0 1 1 1 q n q x~~

~~(137 ) n k m m (k) u n m u [m]x u q 2 S[n, k] x = (−1) e n≥0 k=0 n! m≥0 [m]! m≥0 [m]!~~

~~Denoting Sq (n, k) = S[n, k]; sq (n, k) = s[n, k], and extending the deﬁnitions of Sq and sq for all n ∈ Z (given by (137)) Gould proved also that~~

~~(− − , ) = k ( , ) sq−1 n 1 k q Sq n k ( ) (− − , ) = k ( , ), ∈ Z 138 Sq−1 n 1 k q sq n k n~~

He also discussed the relationship of these numbers to the q-Bernoulli numbers of higher order as deﬁned by L. Carlitz [91], [93]. For certain other q-binomial coef- ﬁcients, connected to the ordinary Stirling numbers, see [199]. −(k) Let S[n, k] = q 2 S[n, k], and the q-exponential polynomial given by

~~n k en[z] = S[n, k]z . k=0~~

In 1999 J. Cigler [109] has studied the determinant of a Hankel matrix, proving that (n+1) (n+1) det(ei+ j [z])0≤i, j≤n = q 3 [0]![1]! ...[n]!z 2 (139) Recently R. Ehrenborg [147] has considered other determinants involving q- Stirling numbers, or ordinary Stirling numbers. For example, he proved that

+ + (k n 1)−(k) 0 1 n det(S[k + i + j, k + j])0≤i, j≤n = q 3 3 [k] [k + 1] ...[k + n] ,(140) where S[k + i + j, k + j] are q-Stirling numbers of the second kind. His methods permits also to show that ( + + , + ) k(n+1) S k i j k j = 2 ,() det ( + + ) ( ) ( + ) ...( + ) 141 k i j ! 0≤i, j≤n 2k !! 2k 2 !! 2k 2n !! where S(k + i + j, k + j) are ordinary Stirling numbers of the ﬁrst kind, and m!! = m(m − 2)...2isthe double-factorial.

~~483 CHAPTER 5~~

ForHankel determinants of Bell polynomials, see P. Delsarte [132], who showed that (142) N N(N+1)/2 det(Bi+ j (x))0≤i, j≤N = k! x k=0 See also A. Junod [238] for more results. For non-central q-factorial coefficients and q-Stirling numbers, see Ch. A. Char- alambides [103]. G. The p, q-Stirling numbers have been introduced by M. Wachs and D. White [469]. The p, q-analogue of nonnegative integers are defined by [0]p,q = 0, n − n p q n−1 n−2 n−1 [n] , = = p + p q + ···+q for n ≥ 1. p q p − q , ∗ , , The signless p q-Stirling numbers of the first kind sp,q [n k] and the p q- Stirling numbers of the second kind Sp,q [n, k] are defined by ∗ , = ∗ − , − + − ∗ − , ( ≥ )() sp,q [n k] sp,q [n 1 k 1] [n 1]p,q sp,q [n 1 k] n 1 143 ∗ , = ∗ , = < > with boundary conditions sp,q [0 0] 1, sp,q [n k] 0 for k 0ork n. Similarly, Sp,q [n, k] are defined by

Sp,q [n, k] = Sp,q [n − 1, k − 1] + [k]p,q Sp,q [n − 1, k] (n ≥ 1), (144) with the analog boundary conditions as above. We note that there exist variations to these coefﬁcients, for example Sp,q [n, k], Sp,q [n, k] deﬁned by (see [375]); (145)

~~k−1 Sp,q [n, k] = q Sp,q [n − 1, k − 1] + [k]p,q Sp,q [n − 1, k],~~

k−1 k−1 Sp,q [n, k] = q Sp,q [n − 1, k − 1] + (p[k − 1]p,q + q )Sp,q [n − 1, k − 1], 2 k−1 where [n]p,q = 1 + pq + (pq) + ···+(pq) . , ∗ Forauniﬁed combinatorial approach for the p q-Stirling numbers sp,q and Sp,q , see A. de Medicis´ and P. Leroux [302], where one can ﬁnd also the following generating functions formula and recurrence relations: ∗ , n−r r = ( + )( + )...( + − ), ( ≥ ), sp,q [n r]y x x x [1]p,q y x [2]p,q y x [n 1]p,q y for n 1 r≥0 (146) r 1 1 1 Sp,q [k + r, k]x = · ... ; − , − , − , r≥0 1 [1]p q x 1 [2]p q x 1 [k]p q x

~~484 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~n ∗ + , + = − ... + ∗ , ; ( ) sp,q [n 1 k 1] [n]p,q [n 1]p,q [ j 1]p,q sp,q [ j k] 147 j=k n + , + = + n− j , ; Sp,q [n 1 k 1] [k 1]p,q Sp,q [ j k] j=k~~

n−k−1 ( − ) , = − − , ( j+1 + ···+ j+1); ( ) n k Sp,q [n k] Sp,q [n 1 j k] [1]p,q [k]p,q 148 j=0 n , = n−k ∗ 1 n m (−1) s [n, k]S , [k, m] = (orthogonality) p,q p q 0, n = m k=m n , = k−m ∗ 1 n m S , [n, k](−1) s [k, m] = p q p,q 0, n = m k=m n ∗ , = (− ) j−k j−k ∗ + , + , sp,q [n k] 1 [n]p,q sp,q [n 1 j 1] j=k

~~n−k j Sp,q [n, k] = (−1) [k +1]p,q [k +2]p,q ...[k + j]p,q Sp,q [n +1, k + j +1],(149) j=0 n (1 − [2]p,q )(1 − [3]p,q )...(1 − [k − 1]p,q )Sp,q [n, k] = n − 1 (n ≥ 2). k=2~~

~~The last identity is similar to an identity for Sq [n, k] due to A. Mercier [305]:~~

n k k−2 (−1) q [k − 2]!Sq [n, k] = n − 1 for n ≥ 2 k=2 Forfar-reaching generalizations and analogs, see A. de Medicis´ and P. Leroux [301]. Let U = (A,w),where A = (ai )i≥0 denotes a strictly increasing sequence of nonnegative integers, and w : N → K denotes a function from N to a ring K . The U- Stirling numbers of the ﬁrst kind (without sign) and of the second kind, are deﬁned by ∗ ∗ ∗ sU (n, k) = sU (n − 1, k − 1) + wn−1sU (n − 1, k), (150)

SU (n, k) = SU (n − 1, k − 1) + wk SU (n − 1, k) ∗ ∗ with initial boundary conditions sU (0, k) = δ0,k = SU (0, k), sU (n, 0) = w w ...w ( , ) = wn U 0 1 n−1, SU n 0 0 .For a general study of -Stirling numbers, see [301]

~~485 CHAPTER 5~~

(where another class of Stirling numbers, namely the partial p, q-Stirling numbers, are studied, too). For q-analogue triangular numbers, which arises in the context of distance geometry, see K. B. Stolarsky [426]. In [427] Stolarsky remarks that

~~1 r s + 1 q + q r s ([n]/n) n−1 = (151) 2~~

1 1 qn − 1 is an identity for n =−2, −1, − , and 2, where [n] = is extended to 2 2 q − 1 all real numbers n, and r = r(n), s = s(n) are the roots (in some order) of the polynomial Pn(z),where

~~2 2 2 2 2Pn(z) = 50(1 + n) z − 30(1 + n)(1 + n )z + (n − 2)(n − 1)(1 + n )~~

~~2 In fact√r(n) → ρ, √s(n) → σ as n → 0, where ρ = φ2/5, σ = φ /5, with 1 + 5 1 − 5 φ = , φ = (φ being the golden number). 2 2 Computer algebra shows that~~

1 r s + 1 q + q r s ([n]/n) n−1 − = (n − 2)(1 + n)(2 + n)(−1 + 2n)(1 + 2n)· 2 (q − 1)6 (q − 1)7 991 + 10n − 9n2 · − + (q − 1)8 + O(q − 1)9 26 · 34 · 53 · 7 27 · 34 · 52 · 7 210 · 35 · 54 · 7 (152) Relation (151) is an identity for only ﬁve values of n listed, but for no other value of n.Wenote that the cases n = 0 and n = 1 the left side must be understood as a limit. This limits are in fact famous means, as the logarithmic and identric means. For these means see e.g. H. Alzer [19], H. Alzer and S. L. Qiu [23], B. C. Carlson [99], E. B. Leach and M. C. Sholander [274], A. P. Pittenger [351], J. Sandor´ [384], [385], E. Neuman and J. Sandor´ [324], [325], K. B. Stolarsky [428], M. K. Vamanamurthy and M. Vuorinen [461], C. E. M. Pearce and J. Pecariˇ c´ [344], J. Sandor´ and Gh. Toader [387], L. Losonczi and Z. Pales´ [290], J. Sandor´ and T. Trif [388], etc. Foranextension of the Gauss q-binomial coefﬁcient to sequences, see W. A. Kimball and W. A. Webb [253]. For an application of q-Stirling numbers in the construction of a q-analogue of power series, see J. Satoh [391]. A q-analogue of Bell numbers has been introduced by S. C. Milne [308], [309]. (m) Let Sq satisfy the recurrence (in fact Sq [n, m] = q 2 Sq [n, m]):

~~m−1 Sq [n + 1, m] = q Sq [n, m − 1] + [m]q Sq [n, m],~~

~~486 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS and put n Bq (n) = Sq [n, m]. m=0~~

~~Then Bq (n) satisﬁes the recurrence relation n n j Bq (n + 1) = q Bq ( j) j=0 j Recently, C. G. Wagner [471] has introduced another q-analogue by~~

∗( ) = n ( ) Bq n q Bq n and studied it particularly for q =−1(B−1(n) arises in fermionic oscillators, see M. n n+1 Schork [396]). One has B−1(n) = (−1) ,ifn ≡ 0 (mod 3); = (−1) ,ifn ≡ 1 ( ) = ≡ ( ) ∗ ( ) =+ ≡ ( ) = mod 3 , and 0, if n 2 mod 3 . Similarly, B−1 n 1, if n 0 mod 3 ; −1, if n ≡ 1 (mod 3); = 0, if n ≡ 2 (mod 4). The generating function of B−1(n) is given by ∞ n ω x 1 −ωx −ω2x B− (n) = e − e , 1 − ω − ω n=0 n! 1 1 √ −1 + i 3 where ω = . 2 Foracommon generalization of binomial coefficients, Stirling numbers, and Gaussian coefficients, see B. Voigt [467]. H. Finally, we mention certain other generalizations of Stirling numbers. B. Richmond and D. Merlini [364] and P. Flajolet and H. Prodinger [161] have defined and studied Stirling numbers of complex argument. Therefore they have given asolution to the problem of R. L. Graham, D. E. Knuth and P. Patashnik [200] which asks for a good generalization of the Stirling numbers of the second kind S(n, k) for complex numbers n and k. See also G. Kemkes et al. [247]. For Stirling numbers of the first kind, see B. Gyires [206]. The number of permutations of n objects having ai cycles of length i is a1 a2 an n!/(1 a1!2 a2! ...n an!) and summing up such expressions in the right way gives n the refined Stirling numbers of the first kind, having mi cy- m0 m1 ...mk−1 cles of length i (mod k) for some given positive integer k. These numbers have been introduced by J. Katriel [244], who considered also refined Stirling numbers n of the second kind, as the number of partitions of n objects into m0 m1 ...mk−1 mi non-empty blocks of cardinality i (mod k).With the help of generating functions, recurrence relations can be derived.

~~487 CHAPTER 5~~

By studying a self-adjoint Legendre differential operator, W. N. Everitt, L. L. Littlejohn and R. Wellman [158] have introduced the Legendre-Stirling numbers PS(n, k), which is the coefficient of xn−k in the Taylor expansion of the rational k function (1 − m(m + 1)x)−1. m=1 L. A. Szekely´ [435] has defined the so-called Holiday numbers of the first and second kind, which provide results analogous to similar results for Stirling numbers of the first and second kind. H. Niederhausen [332] has studied Stirling numbers in the algebra of formal Laurent series.

~~4 Congruences for Stirling and Bell numbers~~

We now study congruences for Stirling numbers. A. Let p be a prime number. Since s(p, 1) = (p − 1)!, by Wilson’s theorem one can write s(p, 1) ≡−1 (mod p)(153) p On the other hand, by s(p, p − 1) =− ≡ 0 (mod p),byinduction, and 2 relation (23) applied to n = p one gets that

s(p, k) ≡ 0 (mod p) for 2 ≤ k ≤ p − 2,(154) where p is a prime, and s(p, k) are Stirling numbers of the ﬁrst kind. See e.g. [116] pp. 54-55 for more details. Another proof of (154) follows by Lagrange’s theorem (x p) = x(x − 1)...(x − p p k p p + 1) ≡ x − x (mod p),By(19), (x p) = s(p, k)x ≡ x − x (mod p) k=0 which by s(p, p) = 1 and (153) gives (154). For this idea of using Lagrange and Wilson theorems for generalized Stirling numbers, see E. T. Bell [37]. Since s∗(n, k) = (−1)k+ns(n, k) (see (18)), so (154) holds true also for s∗(p, k).Foran extension of Fermat’s little theorem, with application to Stirling numbers - congruences, see K. Dilcher [143]. We note that (154) has an important application in the arithmetic properties of generalized harmonic numbers H(n, r) = 1/(n0n1 ...nr ) (where r ≥ n0+n1+···+nr ≤n 0, n ≥ 1are integers), since

~~(r + 1)! H(n, r) = s∗(n + 1, r + 2), (155) n!~~

488 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS see A. Gertsch [182]. Other congruence properties from [182] with applications, are e.g. s∗(p + 1, r + 2) ≡ 0 (mod p2) if 0 < r < p − 3isodd(p = prime); (156) s∗(2p, r + 2) ≡ 0 (mod p2) if 0 < r < p − 1isodd (157) By letting n = p in (16), by (154) one can deduce immediately s∗(p − 1, k) ≡ 1 (mod p)(1 ≤ k ≤ p − 1) Selecting n = p − 1, by the above result, and induction on k we get (see F. T. Howard [220]): s∗(p − 2, k) ≡ 2p−k−1 − 1 (mod p)(0 ≤ k ≤ p − 2) More generally, if p is a prime, h > 0, 0 ≤ m < p, then h h s∗(hp + m, k) = (−1)h−i s∗(m, k − h − i(p − 1)) (mod p) i=0 i A. Nijenhaus and H. S. Wilf [336] have proved that if k + n is odd, then s∗(n, k) is divisible by the odd part of n − 1, and Howard [220] has improved this assertion: if k + n is odd, then n(n − 1) s∗(n, k) ≡ 0 (mod ). 2 A common generalization of Wilson’s theorem, and other congruence properties is s∗(n + p, k) ≡ (p − 1)s∗(n, k − 1) + s∗(n, k − p)(mod p), (158) ∗ see B. E. Sagan [377]. Taking n = 0, k = 1in(158) and recalling that s (0, k) = δ0,k, we can deduce Wilson’s theorem (p − 1)! ≡ p − 1 (mod p) ≡−1 (mod p).Using the fact that s∗(n + 1, 1) = n! ≡ 0 (mod p) for n ≥ p,byinduction on k,from (158) one gets s∗(n, k) ≡ 0 (mod p), if n > kp (159)

Let n = n1 p + n0 (0 ≤ n0 < p), k = k1(p − 1) + k0 (0 ≤ k0 < p − 1) in (158). Then via induction on n one can deduce ∗ k−1 n1 ∗ s (n, n − k) ≡ (−1) s (n0, n0 − k0)(mod p)(160) k1 This implies a result of H. Gupta [205]: ∗ s (np, k) ≡ 0 (mod p) if p − 1 np − k, and n (161) s∗(np, k) ≡ (−1)m (mod p) if m(p − 1) = np − k m

~~489 CHAPTER 5~~

B. The method of proof of (158) is by using Mobius¨ inversion over the lattice of subgroups of a given Abelian group acting on a set. Such an idea originates to J. Peterson [346] who in 1872 used the action of the cyclic group of order p to prove the congruences of Fermat and Wilson. This method has been rediscovered many times (see L. E. Dickson [139]), but a systematic development was still lacking until the papers by G. C. Rota and B. E. Sagan [372] (who investigated congruences from groups acting on functions), J. H. Smith [414] (who has considered wreath products), or I. M. Gessel [186] (congruences for groups acting on graphs). The method can be summarized as follows. Let G be a finite group with identity element e, and let S be a finite set on which G acts. Given s ∈ S, the stabilizer of s is Gs ={g ∈ G : gs = s} and the orbit of s is Os ={t ∈ S : t = gs for some g ∈ G}.Itiswell known that card Os = card G/card Gs.Anelement s ∈ S is said to be aperiodic whenever Gs = e.Bythe above result, s is aperiodic iff card Os = card G,soweimmediately have the following result: The number of aperiodic elements of S is divisible by card G.(162) Theorem (162) is the base of proving many congruences. The only thing needed is the determination of the number of aperiodic elements. Let L(G) denote the lattice of all subgroups H ≤ G ordered by inclusion. For H ∈ L(G) define

~~α(H) = card{s ∈ S : Gs = H} Then (162) can be restated as~~

~~α(e) ≡ 0 (mod |G|), (163) where |G|=card G. Calculating α directly is difﬁcult, but there is another function β which is easier to work with. Let~~

~~β(H) = card{s ∈ S : Gs ⊃ H}.~~

Then clearly β(H) = α(K )(164) K ≥H,K ∈L(G) By using Mobius¨ inversion (see Chapter 2), one obtains µ(H)β(H) ≡ 0 (mod |G|), (165) H∈L(G) where µ is the Mobius¨ function of L(G). In order to obtain congruences from (165) we need only to substitute various groups G and sets S on which they can act.

~~490 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~Forexample, let G = C p the cyclic group of order p (p prime). Then µ(e) = 1, µ(C p) =−1, so (165) becomes~~

~~β(e) ≡ β(C p)(mod p)(166)~~

Let S be the set of all permutations σ of {1, 2,...,n + p} with k cycles, with C p acting by conjugation. Then card S = s∗(n + p, k), and (158) follows from (166). (r) = r µ( ) = (− )r 2 By considering G C p (when G 1 p ), one can deduce an extension of (159): s∗(n, k) ≡ 0 (mod pr ) if n > krp (167) The method of (165) applies also to the Stirling numbers of the second kind. Let S be the set of all partitions π of {1, 2,...,n + p} into k subsets, which is counted by S(n + p, k). The action on C p on subsets extends naturally to such partitions. Then β(e) = card S = S(n + p, k),Tocompute β(C p) there are two cases. If {1} is a singleton block of π then so are {2}, {3},...,{p}. This means that there are S(n, k − p) ways to choose the rest of the blocks of π.If1is in a block of size at least two, then 2, 3,...,p are also in the same block. Thus the set {1, 2,...,p} is acting like a single element, which we shall denote by P,soweare just partitioning the set {P, p + 1,...,p + n} into k blocks. This gives a ﬁnal contribution of S(n + 1, k) to β(C p), and hence (166) implies S(n + p, k) ≡ S(n + 1, k) + S(n, k − p)(mod p)(168)

This can be proved also by induction on n. From (168), via an inductive argument on h, the following can be proved (see [220]): If p is prime and h > 0, 0 ≤ m < p, then h h S(hp + m, k) ≡ S(m + h − i, k − ip)(mod p)(168 ) i=0 i Nijenhuis and Wilf [336] showed that if k + n is odd, the S(n, k) is divisible by the odd part of k, and Howard [220] improved this to the following: if k + n is odd, then k(k + 1) S(n, k) ≡ 0 (mod ) 2 This can be further reﬁned: if n + k is odd, then k2(k + 1) S(n, k) ≡ 0 (mod ), if k|n, 2 and k3(k + 1) S(n, k) ≡ 0 (mod ), if k2|n 2

~~491 CHAPTER 5~~

~~For n = 0(168) gives~~

S(p, 1) ≡ 1 (mod p); (169) S(p, k) ≡ 0 (mod p) for 2 ≤ k ≤ p − 1, in analogy with relation (154) for s(p, k). p p By (169) and x = S(p, k)(x)k one can reobtain (with a new proof) the k=0 p Lagrange congruence (x)p ≡ x − x (mod p). By repeated application of (168), and an inductive argument, one can obtain a result of H. E. Becker and J. Riordan [35]: If j is deﬁned by p j ≤ k < p j+1, then

~~S(n + p j (p − 1), k) ≡ S(n, k)(mod p)~~

~~= r In the case of G C p one can deduce a more complicated recurrence (see also [377]): r i j r j − 1 + δ , (−1)i S(i, j)[−p(p − 1)]i− j i j · i=0 i j=0 l=0 l~~

·S(n + (r − i)p + l, k − ( j − l)p) ≡ 0 (mod pr )(170) The periods of the Stirling numbers S modulo pr can be derived from the recursion above. One has (L. Carlitz [94]): If p j ≤ k < p j+1, then

~~S(n + pr+ j (p − 1), k) ≡ S(n, k)(mod pr )(171)~~

See also H. Tsumura [451]. The minimum period of (S(n, k))n≥0 (mod p) is computed by A. Nijenhuis and H. S. Wilf [336]. They proved that this period is (p − 1)p j if k ≥ p, and l.c.m.[o(1),...,o(k)] for k < p, where j is the greatest integer such that p j < k, and o(i) denotes the multiplicative order of i in the integers mod p (lcm denotes least common multiple). (171 ) A generalization of (169) appears in [372]: If a = [k/p], then

~~S(p j , k) ≡ S(p j−r , k)(mod p j−a−r+1)(172)~~

~~In particular, if 1 < k < p, then~~

~~S(p j , k) ≡ 0 (mod p)(j ≥ 1)(173)~~

~~492 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

C. The method can be applied also to the Bell numbers B(n),deﬁnedby(9).By letting G = C p,andS = the set of all partitions of {1, 2,...,n + p}, one can deduce aresult of J. Touchard [450] from 1933:

~~B(n + p) ≡ B(n + 1) + B(n)(mod p)(174)~~

~~By iteration this implies a result of M. Hall [210]:~~

~~B(n + (p p − 1)/(p − 1)) ≡ B(n)(mod p)(175)~~

~~We note that for n = 0, relation (174) implies~~

~~B(p) ≡ 2 (mod p) for any prime p (176)~~

For generalizations, see Williams [485] or Becker and Riordan [35], who considered the iterated Stirling numbers introduced by E. T. Bell [38], [39]. For s ≥ 0 deﬁne the numbers S(n, k, s) by

~~n S(n, k, 1) = S(n, k); S(n, k, s) = S(n, i, 1)S(i, k, s − 1)(177) i=0~~

~~When s = 0, the equation (177) forces S(n, k, 0) = δn,k.The corresponding Bell numbers are n B(n, s) = S(n, k, s)(178) k=1 Becker and Riordan prove that if pm−1 ≤ s < pm = q, then~~

B(n + pq − 1, s) ≡ B(n, s)(mod p), (179) i.e. B(n, s) has the period pq − 1 modulo p.Forthe period of S(n, k, s) one has: Let q m j−1 s j s qm = p ,where q = p .Ifqm−1 < s ≤ qm, and qm p ≤ k < qm p , then

~~( + j ( − ), , ) ≡ ( , , )( )() S n qm qm 1 k s S n k s mod p 180~~

Forextensions of these results, see L. Carlitz [94]. An extension of Hall’s result (175) is proved by W. F. Lunnon, P. A. B. Pleasants and N. M. Stephens [293]: p p − 1 B n + pr ≡ B(n)(mod pr )(181) p − 1

~~493 CHAPTER 5~~

p p − 1 C. Radoux [360] conjectured that T = is the minimal period of B p − 1 n (mod p),and showed that if the period of the residues is equal to T for a given prime p, then there exists a number c, depending on p,such that

~~B(c + q) ≡ 0 (mod p) for 1 ≤ q < p.~~

See also [357], [358] on arithmetic properties of B(n). Radoux’s conjecture has been veriﬁed for all primes p ≤ 17 (see N. Kahale [239]). In [239], for a given prime p, there is constructed an integer c such that the Bell numbers B(c + 1), B(c + 2), ...,B(c + p − 1)(182) are divisible by p.For maximum zero strings of Bell numbers (mod p), see J. W. Layman [271], and J. W. Layman and C. L. Prather [272]. For generalizations of Carlitz’s and Radoux’s or Layman’s, Layman and Prather’s results, see A. Mazouz [297], [298]. For the p-adic method, see also a paper from 1976 by D. Barsky [33]. For congruences for the so-called Barrucand numbers, having the generating function exp(xex ), see A. Mazouz [299]. The congruence B(mp) ≡ B(n + 1)(mod p)(183) is due to L. Comtet [116]. A. Gertsch and A. M. Robert [184] have proved that if pkn, then B(np) ≡ B(n + 1)(mod pk+1)(184) This was a conjecture of M. Zuber (see [184]). H. Tsumura [451] proved that

~~B(pk) ≡ (B(p) − 1)k + 1 (mod p), for all k ≥ 1. L. Carlitz proved that~~

~~k+1 B(m + nωp) ≡ B(m)(mod p ),~~

~~p−1 with ωp = 1 + p + ···+ p .A.Junod has recently shown that n n B(m + nps) ≡ sn− j B(m + j)(mod pk+1), j=0 j n n B(m + np) ≡ B(m + j)(mod pk+1) j=0 j~~

~~494 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~k where p n and p is odd. When p = 2, one has modulo 2nZ2: ∗ n n n n, if n is odd, and m ≡ 2 + (mod 3) B(m + 2n) − B(m + j) ≡ 4 j=0 j 0, otherwise~~

~~For all these results, see [238]. The Bell polynomials Bn(x) are deﬁned by n n Bn+1(x) = x Bk(x)(n ≥ 0), B0(x) = 1. k=0 k~~

~~Then B(n) = Bn(1).In1976 C. Radoux [359] proved the following congruence: For k ≥ 1andp prime,~~

p pk Bn+pk (x) ≡ Bn+1(x) + (x + ···+x )Bn(x)(mod pA[x]), (185) where A is either the ring Z of integers or the ring Zp of p-adic integers (and f (x) ≡ g(x)(mod pA[x]) means that f (x) = g(x) + ph(x), where h(x) ∈ A[x]-the ring of polynomials with coefﬁcients in A). For a new proof, see [184]. For x = 1 and k = 1, (185) reduces to Touchard’s congruence (174). I. E. Shparlinkij [401] has proved that the congruence

B(n) ≡ λ(mod p)(186) 2p − 1 has for any integer λ,asolution n with 0 ≤ n ≤ . p − 1 A. Gertsch [183] has applied the theory of congruences of Bell numbers in the study of Kurepa’s left-factorial function

~~!p = k p = 0! + 1! + ···+(p − 1)!~~

~~(see [312], ch. XII, and [266]). Gertsch proved that~~

k p ≡ B(p − 1) − 1 (mod p) for an odd prime p.A.Junod [238] has recently shown that Kurepa’s conjecture (i.e. n k p ≡ 0 (mod p))isequivalent to the following: If p n,then p (B(p − 1) − 1) for all odd primes p, and integers n ≥ 1. Foraconnection of Kurepa’s conjecture with Bernoulli numbers, see V. S. Vladimirov [466]. Congruences for Bell polynomials associated to a partition π of monomials have been studied by I. M. Gessel [186]. Associate with each partition π of {1, 2,...,n}

~~495 CHAPTER 5~~

~~π( ) = i1 i2 ... in π a monomial x x1 x2 xn where ih is the number of blocks of size j in . Deﬁne the polynomial Bn(x) by Bn(x) = π(x), π a partition of {1,...,n}.~~

Then ( ) ≡ p ( ) + n ( )( )() Bn+p x x1 Bn x xi+p Bn−i x mod p 187 i i For congruences modulo m (m arbitrary positive integer) we quote the following results: n µ(d)a!S , a ≡ 0 (mod n)(188) d|n d In particular, if a < p, and p is the smallest prime divisor of n, then n µ(d)S , a ≡ 0 (mod n), (189) d|n d for any positive integer a. Similarly, n ϕ(d)a!S , a ≡ 0 (mod n), (190) d|n d for any positive integer a, where µ is the classical Mobius¨ function, while ϕ is the classical Euler totient function. For (189) and (190), see [372]. The following results can be found in [377]:

~~/ n d n/d i n µ(d) S(i, j)di− j S m + − i, k − dj ≡ d|n i=0 i j=0 d~~

≡ 0 (mod n), (191) / n d n/d i n µ(d) S(i, j)di− j B m + − i ≡ 0 (mod n), (192) d|n i=0 i j=0 d / n d n/d i n µ(d) (d − 1)n/d−i s∗(i, j)di− j s∗ m, k − dj − + i ≡ d|n i=0 i j=0 d ≡ 0 (mod n)(193) D. Let P(n, k) = k!S(n, k) denote the number of ”preferential arrangements” of {1, 2,...,n} with k blocks (a preferential arrangement order on the blocks of π).

~~496 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

Thus P(n, k) is the number of surjections {1, 2,...,n}→{1, 2,...,k}.Suchar- rangements were ﬁrst studied by Touchard [450], and rediscovered by O. A. Gross [202] and I. J. Good [195]. P(n, k) has period p − 1 (mod p) for all k, and for k ≥ p we have

~~P(n, k) ≡ 0 (mod p)(194)~~

~~For n ≥ pr−1, P(n, k) has period pr−1(p − 1) and is congruent to 0 for sufﬁ- ciently large k.Infact, Touchard [450] proved that~~

~~P(m + ϕ(n), k) ≡ P(m, k)(mod n), (195)~~

~~r − ≥ { ai 1 = , } = ai for any m max pi : i 1 r ,where n pi is the prime factorization of i=1 n (and ϕ is Euler’s totient). n Let P(n) = P(n, k). Then (see [450]): k=1~~

~~P(m + ϕ(n), k) ≡ P(m, k)(mod n), (196)~~

~~− ≥ { ai 1 = , } for any m max pi : i 1 r . In 1977 A. T. Lundell [292] has studied other divisibility properties of P(n, k) and of~~

~~n,m = g.c.d.{P(n, k) : m ≤ k ≤ n} (197)~~

~~He proved the following congruence: Let p be a prime, n = Ad + r where 0 ≤ r < d = pα(p − 1), α ≥ 0ifp is odd, α ≥ 1ifp is even. If 1 ≤ t ≤ A, then~~

~~− t 1 A A − 1 − i P(n, k) + (− )t (− )i k S(id + r, k) ≡ 1 1 − − ! i=0 i t 1 i k ≡ (−1)k+t (−1)pq (pq)r · q≥0 pq~~

~~− t 1 A A − 1 − i · (− )i (pq)id ( pt (α + ε)) ( ) 1 − − mod 198 i=0 i t 1 i where ε = 1ifp is an odd prime, ε = 2ifp = 2.~~

~~497 CHAPTER 5~~

t The following results give the exact conditions for p to divide n,m for p an odd prime and t < p: Let p be an odd prime, n = A(p − 1) + r, where 0 ≤ r < p − 1, and let i A = a0 + ai p be the p-adic expansion of A with aγ = 0ifA = a0 (see i≥γ Chapter 4). If A = a0, let µ = max{a0, t − γ − 1}. Then for 1 ≤ t < p,

(1) If 1 ≤ t ≤ a0 and t ≤ r, t then p |n,m iff m >(t − 1)(p − 1) + r; (2) If 1 ≤ t ≤ a0 and t > r, t then p |n,m iff m > t(p − 1) + r; (3) If A = a0, and 0 ≤ a0 < t, t then p n,m; ( ) = , < ≤ µ< , − γ ≤ , 4 If A a0 0 a0 t and t r ( ) t 199 then p |n,m iff m >µ(p − 1) + r; (5) If A = a0, 0 < a0 ≤ µµ(p − 1) + r; (7) If A = a0, 0 = a0 ≤ µ t(p − 1) + r.

For the divisibility of n,m by powers of 2, a table is provided in [292]. Let γp(m) be the p-adic order of m (i.e. the highest power of the prime p which divides m). In 1994 T. Lengyel [278] showed that

n γ2(P(2 , k)) = k − 1ifn ≥ l(k), (200) where l(k) is a function depending on k.Fork ≥ 5, l(k) can be chosen so that l(k) ≤ k−2. In [279] Lengyel has shown that the 5-adic analysis of the Fibonacci and q Lucas numbers Fn, Ln plays a major role in determining γ5(P(n, k)) for n = a · 5 , k = 2b · 5z, where a, b, q are positive integers such that (a, 5) = (b, 5) = 1, and 4|a, while z is a nonnegative integer. For instance, when q is sufﬁciently large and z > 0, one has the congruences

~~b ·5z −1 q+1 P(n, k) ≡−2 · 5 2 L · z (mod 5 ), if b is even; b 5 ( ) · z − 201 b 5 1 q+1 P(n, k) ≡ 2 · 5 2 Fb·5z (mod 5 ), if b is odd~~

q Notice that for n = a·5 ,4|a, (a, 5) = 1, and q sufﬁciently large, γ5(P(n, k)) can depend on n only if k is odd. More general results can be found in I. M. Gessel and T. Lengyel [187]. Let n = a(p − 1)pq , where p is a prime, (a, p) = 1, q sufﬁciently

498 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS large, and k/p is not an odd integer. Then k − 1 γ (P(n, k)) = + τ (k), (1 ≤ k ≤ n), (202) p p − 1 p where τp(k) ≥ 0(τp(k) = 0ifk = multiple of (p − 1)). They prove also that for any odd prime p,ifk/p is an odd integer, then

q γp(P(a(p − 1)p , k)) > q (203) k − 1 If q is sufﬁciently large, and k ≡ 5 (mod 10), then γ (P(n, k)) = + 5 4 τ5(k), where τ5(k) = γ5(k + 1) if k ≡ 9 (mod 20); γ5(k) if k ≡ 10 (mod 20); γ5(k + 2) if k ≡ 18 (mod 20);0,otherwise. (204) For τp(k) given by (202), Gessel and Lengyel conjecture that: 1) if k is divisible by 2p,butnot by p(p − 1), then

~~τp(k) = γp(k);~~

~~2) if k + 1isdivisible by 2p,butnot by p(p − 1), then~~

~~τp(k) = γp(k + 1).~~

F. W. Clarke [111] conjectures that if p is an odd prime and n > k, then p divides P(n, k) less than n times. For γ2(S(n, k)) see D. M. Davis [129]. These exponents are important in algebraic topology (see also M. Crabb and K. Knapp [123] and D. M. Davis [130]). Let sp(m) denote the sum of digits in the p-ary representation of m (see Chapter 4 for this function). In [277] T. Lengyel proves that s (n − k) − s (n) − k(p − 2) γ (S(n, n − k)) ≥ p p + 1 (205) p p − 1 for every prime p ≥ 3anintegers 1 ≤ k ≤ n. He proves also that for all 1 ≤ k ≤ 2n,

~~n γ2(S(2 , k)) = s2(k) − 1,(206) and that for all odd a ≥ 3 and all integers 1 ≤ k ≤ n,~~

~~γa(P(n, n − k)) ≥ γa(n!) − k + 1 (207)~~

~~For all a ≥ 3 with γ2(a) ≥ 2, the inequality (207) holds for 1 ≤ k ≤ n.Fork = 1 we have γa(P(n, n − 1)) = γa(n!(n − 1)/2) ≥ γa(n!) − 1.~~

~~499 CHAPTER 5~~

For modiﬁed Stirling numbers divisibility, see J. Gonzalez´ [194]. E. In 1990 F. T. Howard [224] has obtained certain congruences for Stirling numbers, in terms of the 2rth Bernoulli numbers B2r . Let p be an odd prime, and suppose pt n.Assume that 0 < 2r < 2p − 2 and 1 < 2r + 1 < 2p − 2. Then we have n n − 1 s∗(n, n − 2r) ≡− B (mod p2t ), 2r 2r 2r (208) n2(2r + 1) n − 1 s∗(n, n − 2r − 1) ≡− B (mod p3t ), 4r 2r + 1 2r n n + 2r S(n + 2r, n) ≡ B (mod p2t ), 2r 2r 2r (209) n2(2r + 1) n + 2r + 1 S(n + 2r + 1, n) ≡ B (mod p3t ). 4r 2r + 1 2r If n = p and 0 < 2r < p − 1, 1 < 2r + 1 < p − 1, congruences (208) reduces to results of J. W. L. Glaisher, while (209) reduce to results of N. Nielsen. In [224] there are proved also congruences in terms of Bernoulli numbers of higher order. As we have seen in relations (62), (63), the Stirling numbers of the ﬁrst kind s(n, k) are connected to the Bernoulli numbers of order k and degree n by s(k, k − k − 1 k + n n) = B(k).One can deduce similarly that S(k + n, k) = B(−k). n n n n (k) By studying congruence properties of Bn ,A.Adelberg [4], by the application of the above identities, has proved the following congruences for s(l, l − p), resp. S(l + p, l): If p is an odd prime, and l is a positive integer divisible by p, then l − 1 s(l, l − p) ≡ l2/2 (mod pl2)(210) p In particular, γ (s(l, l − p)) ≥ 2γ (l), and p p (211) γp(s(l, −p)) = 2γp(l) if γp(l)>1. Similarly, if p is an odd prime, and p|l, then

~~S(l + p, p) ≡ ((l/p) + 1)l2/2 (mod pl2), (212) so γ (S(l + p, p)) ≥ 2γ (l), and p p (213) γp(S(l + p, p)) = 2γp(l) if γp(l)>1~~

~~500 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

R. Sanchez-Peregrino´ [382] has studied congruences for ”poly-Bernoulli numbers” deﬁned by n (−1)mm! Bk = (−1)n S(n, m). n ( + )k m=0 m 1 ≥ ( − )| + ≤ ≤ + k k ≡− He proved that if k 2 and p 1 n, k 2 p n 1, then p Bn 1 ( ) k mod p for primes p.For the numbers Bn ,seeM. Kaneko [240]. F. In 1963 L. Carlitz [88] has considered congruences for S(n, k) for k ≡ r (mod m).For example n − m − 1 S(m, 2m) ≡ (mod 2); m − 1 (214) n − m − 1 S(n, 2m + 1) ≡ (mod 2), m l S(n, 3m) ≡ (mod 3)(m = n − 2l − 2), m − 1 l S(n, 3m + 1) ≡ (mod 3)(m = 2n − 2l − 1orn − 2l − 2), (215) m l S(n, 3m + 2) ≡ (mod 3)(m = n − 2l − 2) m

~~Let θ0(n) denote the number of coefﬁcients S(n, 2m) that are odd, and θ1(n) denote the number of coefﬁcients S(n, 2m + 1) that are odd. Then by (214) one can write~~

~~θ1(n) = θ0(n + 1) and θ0(2n) = θ0(n) + θ0(n − 1), θ0(2n + 1) = θ0(n + 1).~~

~~Let ω0(n) = θ0(n + 2) = number of coefﬁcients S(n + 2, 2m) that are odd. Then we have~~

~~ω0(2n) = ω0(n) + ω0(n − 1), ω0(2n + 1) = ω0(n)(216)~~

~~Among the properties of ω0 we mention the following:~~

~~r r ω0(2 ) = r + 1,ω0(2 s − 1) = ω0(s − 1)(r ≥ 0, s ≥ 1).~~

~~r0 r0+r1 r0+r1+···+rk If n = 2 + 2 + ···+2 (r0 ≥ 1,...,rk ≥ 1) and~~

~~r j r j +r j+1 r j +···+rk n j = 2 + 2 + ···+2 (0 ≤ j ≤ k)(i.e. n0 = n),~~

501 CHAPTER 5 then ω0(n j ) = (1 + r j )ω0(n j+1) − ω0(n j+2)(0 ≤ j < k)(217) where nk+1 = 0. When r0 = r1 = ···=rk = r we get εk+2 − ε−k−2 ω0(n) = (r = 1), (218) ε − ε−1 1 where ε = {1 + r + (1 + r)2 − 4} and n = 2r (1 + 2r + 22r +···+2kr). 2 k For r = 1, we have ω0(2(2 − 1)) = k + 1. For upper bounds on ω0(n) we quote

~~ω0(n) ≤ (1 + r0)(1 + r1)...(1 + rk) and log n (219) ω (n)< 2 , 0 k + 1 where the last one is optimal for r0 = r1 = ···=rk = r. Put S(x) = ω0(n). 0≤n≤x Then~~

S(2n + 1) = 2S(n) + S(n − 1), S(2n) = S(n) + 2S(n − 1)(220) 1 Since S(2r − 1) = (3r + 1) and S(3 · 2r ) = 3r+1 + 2r + 2, by putting λ = 2 log 3/ log 2, it follows that S(2r − 1) 1 S(3 · 2r ) lim = , lim = 31−λ,(221) r→∞ (2r − 1)λ 2 r→∞ (3 · 2r )λ S(n) S(n) so lim inf < lim sup , proved by other methods by P. T. Bateman, too (see →∞ λ λ n n n→∞ n [88]). For the distribution of Stirling (and Euler) numbers mod p in arithmetical triangles, see B. A. Bondarenko et al. [55]. For a given prime p,and positive integer N, they examine the number of elements of Stirling number having residues mod p equal to k for k ≤ p − 1inboth row N and the triangle with base N in the table of these coefficients. G. The associated Stirling numbers of the first kind d(n, k), and of the second kind b(n, k) (see [116]) are defined by k n s∗(n, n − k) = d( k − j, k − j) , 2 − j=0 2k j

502 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS and k n S(n, n − k) = b( k − j, k − j) 2 − j=0 2k j respectively. These numbers can be deﬁned also by means of the generating functions ∞ (− log(1 − x) − x)k = k! d(n, k)xn/n!, n=2k ∞ (ex − x − 1)k = k! b(n, k)xn/n! n=2k These imply at once that

~~d(n, k) = (n − 1)d(n − 1, k) + (n − 1)d(n − 2, k − 1), with d(0, 0) = 1, d(n, 0) = 0 (n ≥ 1); d(n, 1) = (n − 1)! (n > 1); d(n, k) = 0 (n < 2k or k < 0), and~~

b(n, k) = kb(n − 1, k) + (n − 1)b(n − 2, k − 1) with b(0, 0) = 1; b(n, 0) = 0 (n ≥ 1); b(n, 1) = 1 (n > 1); b(n, k) = 0(n < 2k or k < 0). The numbers d(n, k) and b(n, k) are connected with s∗(n, k) and S(n, k) by the following formulae: k n d(n, k) = (−1) j s∗(n − j, k − j), j=0 j k n b(n, k) = (−1) j S(n − j, k − j) j=0 j The above relations imply at once that

~~d(n, k) ≡ 0 (mod (n − 1)) (n ≥ 2)~~

~~d(p, k) ≡ 0 (mod p)(k ≥ 2), p prime; b(p, k) ≡ 0 (mod p), b(p + 1, k) ≡ 0 (mod p)(k ≥ 2) More generally, one has (see Howard [220]): If h > 0, 0 ≤ m < p, then~~

~~d(ph + m, k) ≡ (−1)hd(m, k − h)(mod p),~~

503 CHAPTER 5 and h h b(ph + m, k) ≡ b(m + h − r, k − r)(mod p) r=0 r Thus, b(ph + m, k) ≡ 0 (mod p) if m + h < k or m + h = k and m > 0, b(kp, k) ≡ 1 (mod p). On the other hand, if n − k = (p − 1)w, then w − 1 b(n, k) ≡ (mod p); k − 1 and if n − k = (p − 1)w + v(1 ≤ v ≤ p − 2), then p−1−v w b(n, k) ≡ b(m + v,m)( p) − mod m=0 k m p − 1 If v ≤ , the upper limit of summation can be replaced by v. 2 H. We now study congruences for q-Stirling numbers and related q-coefﬁcients (see (135)-(136) for deﬁnitions). For the q-Stirling numbers of the second kind, B. E. Sagan [376] proved that if k ≡ k0 (mod d), then

S[n, k] ≡ S[n − 1, k − 1] + [k0]S[n − 1, k] (mod [d]), (222) where as usual, [d] = 1 + q + q2 + ···+qd−1.More generally, n S[n, k] = S[m − 1, d − 1]S[n − m, k − d] (mod [d])(223) m=d for all n ≥ d. The generating function of S[n, k]isgivenby xk S n, k xn = , [ ] ( − )( − )...( − ) n≥0 1 x 1 [2]x 1 [k]x and one has the following congruence: If k = k1d + k0,where 0 ≤ k0 < d, then xk S[n, k]xn ≡ (mod [d]) {( − )...( − − )}k1 ( − )...( − ) n≥0 1 x 1 [d 1]x 1 x 1 [k0]x (224)

~~504 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

For the case d = 2 one can write: ∗ n − k − 1 S[n, k] ≡ 2 (mod [2]), (225) n − k ∗ k k where is the integer part of . 2 2 Related to the periods of S[n, k], one has the following analogue of the Nigenhuis-Wilf result for S(n, k) (see (171 )). Let p be a prime. Then the sequence

(S[n, k])n≥0 (mod [p])(or (mod p)) has minimum period 1 if k = 0, 1; (p − 1)p j if k ≥ p; p · lcm[0(1),...,0(k)], if 1 < k < p, where j is the least integer such that p j ≥ k.(226) The sequence (S[n, k])n≥0 (mod [d]) is not periodic unless k = 0, 1ord = 2, 3 and k ≤ d.Inthese cases the minimum period is 1 if k = 0, 1ork = d = 2; 6, if k = 2, 3 and d = 3. (227) For the q-Stirling numbers s∗[n, k] the analogs of relations (222), (225) are the following: If n − 1 ≡ m (mod d), then s∗[n, k] ≡ s∗[n − 1, k − 1] + [m]s∗[n − 1, k] (mod [d]), (228) and s∗[n, k] ≡ s∗[n − 2, k − 2] + s∗[n − 2, k − 1] (mod [2])(229) The generating function of s∗[n, k]isgivenby s∗[n, k]xk = x(x + [1])(x + [2])...(x + [n − 1]), k≥0 and reducing this equation modulo [2] and taking the coefﬁcients of xk,weget: n ∗ s∗[n, k] ≡ 2 (mod [2]), (230) n − k n ∗ n where is the integer part of .As(225), this relation can be proved by combi- 2 2 natorial arguments, too. The coefﬁcients s∗[n, k] are periodic in a trivial sort of way: For any d ≥ 1, s∗[n, k] ≡ 0 (mod [d])(231) for n ≥ kd + 1.

~~505 CHAPTER 5~~

This follows by induction on k and (135 ) applied to n = kd.For proofs of (224)-(231), see Sagan [376]. For the q-binomial coefficients of Gauss, one has ([376]): n n − d n − d ≡ + (mod φ (q)) (232) k k − d k d for all n ≥ d, where φd is the dth cyclotomic polynomial (see Chapter 3). The q-analogue of the famous Lucas congruence theorem for binomial coefficients (see Chapter 2) was first published in a paper by G. Olive [342] (and perhaps was known to Gauss) and rediscovered by J. Desarm´ enien´ [133], A. Blass [50], V.Strehl [430]. A different q-analogue is given in R. D. Fray [166]. The proofs from [342]-[50], [166] were all algebraic, the one from [430] uses combinatorial arguments. The result can be stated as follows: Let n = n1d + n0, k = k1d + k0, with 0 ≤ n0, k0 < d. Then n n1 n0 ≡ (mod φd (q)) (233) k k1 k0 ForaLucas-type congruence for S(n, k),seeR. Sanchez-Peregrino´ [382]. The q-multinomial coefficients can be defined as follows. Let k1,...,kl be nonnegative integer, and put n = k1 + k2 + ···+kl . Then − − − − ···− n = n n k1 ... n k1 k2 kl k1, k2,...,kl k1 k2 kl n For n = k1 + ···+kl , define = 0. Then (232) and (233) have the k1,...,kl following extensions to these coefficients (see [376]): n l n − d ≡ ( φ (q)), ( ) ,..., ,..., − ,..., mod d 234 k1 kl i=1 k1 ki d kl for all n ≥ d, and

~~if n = n2d + n0, ki = ki1 + ki0, with 0 ≤ n0, ki0 < d for 1 ≤ i ≤ l.~~

~~Then n nl n0 ≡ (mod φd (q)) (235) k1,...,kl k11, k21,...,kl1 k10, k20,...,kl0 n where l is the ordinary multinomial coefﬁcient. k11,...,kl1~~

~~506 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

For Lucas’ theorem for generalized binomial coefﬁcients, see D. Wells [482], D. F. Bailey [30]. See also Kimball and Webb [253]. For congruences involving the number of involutions of permutations, see S. Chowla, I. N. Herstein and W. K. Moore [106], S. Chowla, I. N. Herstein and W. R. Scott [107], L. Moser and M. Wyman [316], B. E. Sagan [377]. For derangements and cycle indicators, see J. Riordan [367], I. M. Gessel [186] and B. E. Sagan [377]. For combinatorial properties and applications, see also L. Comtet [116].

5 Diophantine results In this section we shall study some Diophantine results on the Stirling numbers of the second kind S(n, k).In1989 A. Pinter´ [348] (published in 1992) has shown that if a ≥ 1isaninteger, and if S(n, n −a) is an mth power for some n > a, m ≥ 3, then n < C1 = C1(a), where C1 is an effectively computable positive constant depending on a.(236) The proof is based among others on results of A. Baker [32] on the solutions of hyperelliptic equations, and A. Schinzel and R. Tijdemans’s [393] results on the equation ym = P(x). Theorem (236) implies a result of R. Tijdeman [444] which says that there are effectively computable upper bounds for the solutions of equation n + a = ym (a > 1, m > 1, n ≥ 1, y > 1 integers) a with a = 2, m ≥ 3 and a = 3, m ≥ 2. See also K. Gyory¨ [207], [208], for equations of this type. Further A. Pinter´ [349] has shown that for ﬁxed 1 < a < b the solutions of the equation S(m, a) = S(n, b) satisfy max(m, n) 0isaconstant. B. Brindza and A. Pinter´ [62] have studied equations of type

S(x, x − a) = byz,(238) and S(x, a) = byz (239) For equation (238) they proved that if x > 2b · 16a · a8a,andy > 1, then z(7.5 + log z)−2 < 11000(log b + 8a log a + 3a). The same bound is valid in the case of s∗(n, k) (signless Stirling numbers of the ﬁrst kind).

~~507 CHAPTER 5~~

Forequation (239) they proved that if y > 1 (with a > 1), then z is bounded by an effectively computable constant in terms of a and b. Further, if z ≥ 3isfixed, or z = 2 and a = 1, 3 then max(x, y) can be effectively bounded in terms of a and b. The main tool in the proofs is the theory of linear forms in logarithms. Recently, M. Klazar and F. Luca [254] have considered more general Diophantine problems. Let d > 1beaninteger, and let P, Q, R ∈ Q[X1,...,Xm]benonconstant polynomials such that P does not divide Q and R is not a dth power. Let Si = S(n, ki ). Then, if k1,...,km are any sufficiently large distinct positive integers, then Q(S ,...,S ) 1 m ∈ Z (240) P(S1,...,Sm) for only finitely many positive integers n. Further,

d R(S1,...,Sm) = x (241) for on;y ﬁnitely many pairs (x, n) ∈ N2. The proofs of those results rely primaly on ﬁniteness results of P. Corvaja and U. Zannier [122], which in turn are based on Schmidt’s subspace theorem, and hence are innefective in terms of bounding the size of solutions.

6 Inequalities and estimates Inequalities for Stirling numbers have been considered earlier, too (see e.g. relation (80) for the r-Stirling numbers, or (76)-(79) for the Stirling polynomials of the second kind. A. L. H. Harper [213] and E. H. Lieb [287] established the strong log-concavity of the sequence (S(n, k))k (see relation (87)); in fact they proved the more precise estimate k − 1 S(n, k − 1)S(n, k + 1) ≤ S2(n, k)(242) k

The strong log-concavity of (S(n, k))n is due to E. Neuman [323], see relation (86). He also showed the log-concavity of (R(n, k, x))n, see relation (77). B. C. Rennie and A. J. Dobson [362] have proved by elementary arguments, that 1 1 n (r 2 + r + 2)r n−r−1 − 1 ≤ S(n, r) ≤ r n−r (243) 2 2 r D. C. Kurtz [267] deﬁned the generalized Stirling numbers of the second kind (A(n, k)) by A(n, k) = A(n − 1, k − 1) + f (k)A(n − 1, k); A(1, 1) = 1, A(1, k) = 0fork > 1, A(n, 0) = 0forn ≥ 1,

508 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS where 1 = f (1) ≤ f (2) ≤ f (3) ≤ ... is an increasing arithmetic function. He showed that n ( f (k))n−k ≤ A(n, k) ≤ ( f (k))n−k (1 ≤ k ≤ n, n ≥ 1)(244) k

Foramore general recurrence, see J. B. Kim and Y. M. Lee [251], where it is shown also that expressions like s∗(n, k)r n−k and S(n, k)r n−k are strong log-concave ∗ (with respect to k). The sequence (s (n, k))k is strong log-concave. In fact a more precise result is due to E. Neuman [321]:

n − k + 1 k s∗(n, k − 1)s∗(n, k + 1) ≤ s∗(n, k), (245) n − k k + 1 for 1 ≤ k ≤ n.The proof is based on the theory of B-splines. Neuman proves also the following relations p(n − k) + k n p S p(n, k) ≤ S(p(n − k) + k, k), (246) k k for p = 0, 1,...;0≤ k ≤ n; where equality holds when p = 0, 1 and for arbitrary value of p in the case n = k. If 0 ≤ k ≤ n, then the following inequalities are also true:

(n + 1 − k)(n + 2 − k) (n + 3 − k)(n + 4 − k) S2(n + 2, k) ≤ S(n, k)S(n + 4, k) (n + 1)(n + 2) (n + 3)(n + 4) (247) and n + 1 − k n + 2 − k S2(n + 1, k) ≤ S(n, k)S(n + 2, k)(248) n + 1 n + 2 J. C. Ahuja [13] deﬁned numbers C(m, k) and D(m, k) by

− + 2mk 1 2m − k C(m, k) = (−1) j+k s∗( j, j − m) j=m+1 j with a similar expression for D(m, k) in which s∗( j, j − m) are replaced by S( j, j − m).These numbers are suitably deﬁned for k = 0, m ≤ k + 1 and (m, k) = (1, 0). It is shown that these numbers are strong log-concave with respect to k.(249) B. In [291] L. Lovasz´ and K. Vesztergombi have studied the sum

~~k 2 Sk = (m!S(k, m)) m=1~~

~~509 CHAPTER 5 and proved that for large k~~

~~2 2k 2 k Sk ∼ (k!) /(2π(1 − log 2))(log 2) ≈ 0.72019(k!) · (2.081) (250)~~

The numbers m!S(k, m) (denoted by P(k, m) in (194)) for m = 1, 2,...,k, are distributed asymptotically normal as k →∞.(251) n n The numbers P(n) = n!S(k, n) = P(k, n) (studied ﬁrst by Touchard, k=1 k=1 see Part D of 5.24) occur also in the study of the following question: Let k ≥ 1be k k k an integer, and put S0,n = 1 + 2 + ···+n , S1,n = S0,1 + S0,2 + ···+ S0,n,..., Si+1,n = Si,1 + Si,2 + ···+Si,n. Then √ 2n + 1 P(k) + 1 lim Sn,n = √ ; (251 ) n→∞ 4n+1 2π see [294]. T. Popoviciu conjectured the existence of an odd integer λ, with the limit λ being equal to √ .Itiseasy to see that λ = P(k) + 1isodd. π 2 ∗ The sequence (s (n, k))k is unimodal in k. Let el = el (n) denote the sum of the products of the ﬁrst n positive integers ∗ (i.e. the lth elementary symmetric function). Then el (n) = s (n + 1, n + 1 − l) (see e.g. [116]). By the unimodality, it follows that there exists a value k = f (n) which maximizes ek(n); i.e.

~~e1(n)e f (n)+1(n)>···> en(n) = n!~~

~~In 1951 J. M. Hammersley [211] proved that~~

~~e f (n)−1(n) e5, then ∗ 1 log n − ≤ n − f (n) ≤ [log n]∗ (253 ) 2~~

~~510 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

Hammersley conjectured that (252) holds true for all n ≥ 1, i.e. the value of k for which ek(n) is maximal, for a given n is unique. This was proved in 1953 by P. Erdos¨ [153]. The unimodality of (S(n, k))k was proved by L. H. Harper [213] who proved that there are at most two consecutive values of Kn with S(n + k) ≤ S(n + Kn) for all k. He showed also that n K ∼ as n →∞ (254) n log n

~~In 1978 E. R. Canﬁeld [74] proved that if n is sufﬁciently large, and k0 = k0(n) is the solution of the equation~~

~~k0(k0 + 2) ln(k0 + 2)/(k0 + 1) = n,~~

~~∗ ∗ the either [k0] or [k0 +1] , and possible both, yield(s) Kn with S(n +k) ≤ S(n + Kn) for all k. The technique used involves the relation~~

S(n, k) = p(n, k)(exp(r) − 1)kn!/(r nk!), (255) in which p(n, k) is the probability that X1 + ···+ Xk = n, where X1,...,Xk are independent random variables, each taking the value i with probability (exp(r) − −1 i 1) r /i!fori ≥ 1, and the parameter r is chosen to be ln(k0 + 2).Bymeans of asymptotic analysis involving the characteristic function ϕ(ζ) = E{exp(iζ(X −µ))}, an accurate approximation for p(n, k) can be obtained, valid for m − 1 ≤ k ≤ m + 2, ∗ where m = [k0] .Using this approximation it follows S(n, m)>S(n, m − 1) and S(n, m + 2)

~~O(xc) for some c < 1 (256)~~

C. A general theorem of log-convexity and concavity is due to E. A. Bender and E. R. Canﬁeld [41]: If 1, Z1, Z2,... is a log-convex sequence of nonnegative integers, and the sequence (a(n)) is deﬁned by ∞ ∞ a(n) Z xn = exp j x j , n=0 n! j=0 j a(n) then the sequence is log-concave, and (a(n)) is log-convex. (257) n!

~~511 CHAPTER 5~~

B(n) Particularly, the Bell sequence (B(n)) ≥ is log-convex, while is log- n 0 n! concave. (258) The weaker result that (B(n)) is convex can be found e.g. in [116]. More generally, N. Asai et al. [28] have deﬁned the Bell numbers of order k,by b (n) B(k)(n) = k ,(n ≥ ), ( ) 0 expk 0 where ( ) = ( (...( ( )))), expk x exp exp exp x ! ktimes and bk(n) are given by ∞ b (n) ( ) = k n. expk x x n=0 n! ( ) = Since exp2 0 e, one gets

∞ (2) x B (n) ee −1 = xn, n=0 n! i.e. B(2)(n) ≡ B(n). ( (k)( )) By applying (257), Asai et al. [28] proved that the sequence B n n≥0 is log- B(k)(n) convex, is log-concave. (259) n! n≥0 This is an extension of (258) to k ≥ 2. Moreover, the following inequalities hold true: n + m B(k)(n)B(k)(m) ≤ B(k)(n + m) ≤ B(k)(n)B(k)(m)(m, n ≥ 0)(260) n

In 1994 K. Engel [152] conjectured that the sequence (xn)n≥0 given by B(n + 1) x = − 1, n B(n) which is the average number of blocks in a partition of an n-set, is concave. E. R. Canﬁeld [76] proved the conjecture for all sufﬁciently large n.(261) In the proof, the asymptotic of the sequence (B(n)) is used. Namely,

~~− 1 −1 B(n) ∼ (R + 1) 2 exp{n(R + R − 1) − 1},(262) where R · eR = n →∞. This is due to L. Moser and M. Wyman [317].~~

~~512 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~Other asymptotic relations for B(n) can be found in A. M. Odlyzko [339]:~~

~~B(n) ∼ n!(2π R2 exp(R))−1/2 exp(exp(R) − 1)R−n; (262 ) or if n = w log(w + 1), then~~

B(n) ∼ (log w)1/2wn−www as n →∞ (262 ) n n Here w = + O . More precise estimate is log n (log n)2 n n log log n n(log log n)2 n log log n w = + + + O . log n (log n)2 (log n)3 (log n)3 The procedure can be iterated indeﬁnitely to obtain expressions for w with error terms O(n(log n)−α) for as large a value of α as desired. B(n) occurs also in the asymptotic expansion of the number an of Boolean sublat- tices of the Boolean lattice of subsets of {1, 2,...,n} (having the generating function zn an = exp(2z + exp(z) − 1)). One has n≥0 n! n 2 a ∼ B(n) as n →∞, n log n see [339], and the References therein. M. A. Khan and Y. H. H. Kwong [248] have determined the number f (N, r) of pairwise disjoint ﬁnite sequences of rolling an N-faced dice in which the ith sequence consists of ri trials. This is given by

m ∗ N! ( , ) = r j ( , ), ( ) f N r t j S ri t j 263 t j ! i=1,i= j where r = (r1,...,rm), and the sum is taken over t1,...,tm ≥ 1 with t1 + ···+tm = N.Form = 2, r1 = p, r2 = q,itisshown that f (N, p, q) attains its minimal value over all p, q such that p + q = C and |p − q|≤1. (264) D. The convexity or unimodality properties of sequences appear in many ﬁelds of mathematics, see e.g. the survey article of R. P. Stanley [419]. The q-analog of log-concavity has been deﬁned by Butler and Stanley (see B. E. Sagan [378]). The q- analog of the relation ≤ will be denoted by ≤q .Given two polynomials f (q), g(q) ∈ R[q], we write

~~+ f (q) ≤q g(q) if and only if g(q) − f (q) ∈ R [q].~~

~~513 CHAPTER 5~~

Equivalently, for each k ≥ 0, the coefﬁcient of qk in f (q) is less than or equal to the corresponding coefﬁcient in g(q).(This is a partial order on R[q]). A sequence ( fk(q))k≥0 is said to be q-log-concave if

~~2 fk−1(q) fk+1(q) ≤q ( fk(q)) for all k > 0, and it is strongly q-log-concave if~~

~~fk−1(q) fk+1(q) ≤q fk(q) fl (q) for all 0 < k ≤ l.~~

n−1 Forexample, if [n] = [n]q = 1 + q + ···+q , the the sequence ([n])n≥0 is strongly q-log-concave. L. M. Butler [67] and C. Krattenthaler [262] proved that the q-binomial coefﬁcients + n , n , n j ≥ ≥ + k k 0 k n 0 k j j≥0 are strongly q-log-concave. (265) P. Leroux [284] and B. E. Sagan [379], [378] proved that the q-Stirling sequences

∗ ∗ (s [n, k])k≥0,(S[n, k])k≥0,(S[n, k])k≥0,(s [n + j, k + j]) j≥0,(S[n + j, k + j]) j≥0 are all strongly q-log-concave. (266) In [67], [262] the above results were proved by an inductive argument. All the theorems are proved via combinatoric methods in [378]. This method was used also in [284]. The notions of q-log-concavity can be extended to sequences of polynomials fk(q) in N[q], when k ∈ Z. The sequence

~~( fk(q))k∈Z = ..., f−2(q), f−1(q), f0(q), f1(q), f2(q),...~~

2 is q-log-concave if fk−1(q) fk+1(q) ≤q ( fk(q)) and strongly q-log-concave if fk−1(q) fl+1(q) ≤q fk(q) fl (q) for l ≥ k. The sequence q2, q + q2, 1 + 2q + q2, 4 + 2q + q2 is q-log-concave, but not strongly q-log-concave, In [379] it is shown e.g. that for ﬁxed n ≥ 0, the sequences n ∗ ,(S[n, k])k∈Z,(s (n, k))k∈Z k k∈Z are strongly q-log-concave. (267)

~~514 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

As a corollary of a more general result, in [379] appears also: For ﬁxed k ≥ 0, the sequences n ∗ ,(s n, n − k ) ≥ ,(S n, n − k ) ≥ − [ ] n 0 [ ] n 0 n k n≥0 are strongly q-log-concave. (268) ∗ As we can remark, the sequence (s [n, k])n≥0 is missing from the q-log-concavity results. It is absent because it is not q-log-concave even in the case q = 1(take e.g. k = 1). P. Leroux [284] conjectured that for this sequence an ”almost” log-concavity holds true, namely

~~s∗[n − 1, k]s∗[n + 1, k] ≤ [2](s∗(n, k))2 (269)~~

This has been proved in the case q = 1byA.deMedicis´ [300], and by L. Habsieger [209] in the general case (even in a more general setting of sequences). Foranotion and results on p, q-log-concavity, see [284], [379]:

~~The p, q-Stirling numbers Sp,q [n, k] (see (150 )) are p, q-log-concave in n, i.e. for n ≥ 1 and 2 ≤ k ≤ n − 1,~~

2 Sp,q [n, k − 1]Sp,q [n, k + 1] ≤p,q (Sp,q [n, k]) ,(269 ) where f ≤p,q g is to be interpreted as ”g − f is polynomial in the variables p and q with non-negative coefﬁcients”. More generally, for n ≥ 1, 2 ≤ k ≤ l ≤ n − 1we have

Sp,q [n, k − 1]Sp,q [n, l + 1] ≤p,q Sp,q [n, k]Sp,q [n, l] (269 ) ( ) ∗ Relation 269 holds true also when Sp,q is replaced with sp,q . The p, q-unimodality in k if false even for p = 1, since the q-Stirling numbers s∗[n, k] and S[n, k]fail to be q-unimodal in k. E. The asymptotic development of the Stirling numbers of the ﬁrst kind s(n, m) originates from Ch. Jordan [237] and L. Moser and M. Wyman [318]. Jordan proved that for m ﬁxed one has (1 ≤ m ≤ n):

s∗(n, m) ∼ (n − 1)!(log n + γ)n−1/(m − 1)!, as n →∞,(270) where γ is Euler’s constant. Moser and Wyman proved that (270) is valid also for m = o(log n). If m, n →∞such that m ≤ n − O(nα), α ﬁxed, 0 <α<1, then √ s(n, m) ∼ (−1)n+m(n + R)/Rm(R) 2π H,(271)

~~515 CHAPTER 5 where R is the unique solution of~~

~~− n 1 R = m, + k=0 R k and − n 1 R2 H = m − ( + )2 k=0 R k √ Finally, if n − o( n) ≤ m ≤ n, then~~

− n 1 n m s(n, m) ∼ (−1)n+m m (272) m 2 In the proof, the saddle point method is used. Another procedure, which is probabilistic,was developed by E. A. Bender [40] to obtain the asymptotic behavior of general numbers a(n, k), with special applications to asymptotic behavior of the Stirling numbers. Previously L. H. Harper [213] had shown that a certain normalized sum of the S(n, k) tends to a Gaussian integral as n →∞.Ch. Knessl and J. B. Keller [255] have used asymptotic methods of applied mathematics such as the ray method and the method of matched asymptotic expansions, to obtain asymptotic expansions for n 1. In this case, (n − 1)! (−1)n−ms(n, m) ∼ (log n)m−1 if n = O(1); (m − 1)!

− n2 n m 1 ∼ if n − m = O(1); (273) 2 (n − m)! − / m − n mn 1 2 n ∼ A(β) + exp (n − m)(log β − 1) + n log 1 + β β2 β m if 0 < < 1, n where β is the unique solution of em/β = 1 + n/β, and the function A(β) has the limiting forms

~~A(β) ∼ 1 for β 1; A(β) ∼ (2πβ)−1/2 for β 1 √ D. Gardy [174] has proved by the saddle point method that e.g. if n = m+o( m), then n m n−m s∗(n, m) = (1 + o(1)) (274) m 2~~

516 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS and if n − m →∞with n − m = o(m2/3), then − n m n m 2 s∗(n, m) = e(n−m) /6m(1 + o(1)) (275) m 2 In 1993 N. M. Temme [439] proved that n s∗(n, m) ∼ eB g(t ) ,(276) 1 m holds uniformly for 0 < m ≤ n as n →∞. Here

1 1/2 B = ψ(u1) − n log(1 + t1) + m log t1, g(t1) = [m(n − m)/nψ (u1)] u1 m with t = and u the unique positive solution of ψ (u) = 0, where ψ is given 1 n − m 1 by ψ(u) = log[(u + 1)(u + 2)...(u + n)] − m log u. H. S. Wilf [484] provided a considerably more explicit formula when m = O(1), m ≥ 1. He proved that for any ﬁxed m we have 1 (log n)m−1 (log n)m−2 (log n)m−2 s∗(n, m) = γ + γ + ···+γ + O , (n − 1)! 1 (m − 1)! 2 (m − 2)! m n (277) where the γ j are the coefﬁcients in the expansion

~~∞ 1 (6γ 2 − π 2)z3 2γ 3 − γπ2 + 4ζ(3) = γ z j = z + γ z2 + + z4 + ... ( ) j z j=1 12 12~~

~~(γ is Euler’s constant). In fact the coefﬁcients (γn) satisfy the recurrence " n−2 1 n− j−1 γn+1 = γ · γn + (−1) ζ(n − j)γ j+1 (n ≥ 1,γ1 = 1). n j=0~~

The asymptotic sign fluctuations of the sequence (γn) have been studied by W. K. Hayman [215]. See also M. A. Evgrafov [159] (see pp. 92-94). Wilf finds for fixed m also the complete asymptotic series of the Stirling numbers of the first kind, by showing that this is obtained by equating the coefficients of xm on both sides of the relation " ∞ 1 ex log n s∗(n + , m + )xm ≈ ψ (x)/n j ,() 1 1 ( ) exp j 278 n! m xr x j=1

~~517 CHAPTER 5~~

where the polynomials ψ j (x) are given by j+1 j s (−x) j − 1 B + − (−x) ψ (x) = + j 1 s ( j = , ,...) j ( + ) − ( + − ) 1 2 j j 1 s=1 s 1 s j 1 s with Bk being Bernoulli numbers. This provides improvements of relation (277). For example, with error O(n−3) one ﬁnds that k r 1 ∗ (log n) 1 s (n + 1, m + 1) = γk+1−r + (γk−r + γk−r−1)+ m! r=0 r! 2n 1 −3+ε + (−2γ − − 3γ − − + 2γ − − + 3γ − − ) + O(n )(279) 24n2 k r k r 1 k r 2 k r 3 H.-K. Hwang [230] has proved that formula (277) holds for m in the range 1 ≤ (log n)m m ≤ θ log n for any θ>0, if the error term is replaced by O . More m!n generally, he showed that for any θ>0andk ∈ N, one has asymptotically ∗ k m s (n, m) , (log n) (log n) = m r + O ,(n →∞)(280) ( − ) r k+2 n 1 ! r=0 n m!n uniformly for 1 ≤ m ≤ θ log n, where m,r (X) are certain polynomials in X of degree m − 1. One may roughly interpret theorem (280) as saying that the Stirling numbers s∗(n, m) are asymptotically Poisson distributed of parameter (log n). This certainly implies the asymptotic normality of s∗(n, m)/n!inthe sense of convergence in distribution. A more convenient uniform asymptotic expansion is the following (see [230]): ∗ s (n, m) satisfy, uniformly with respect to m,2 ≤ m ≤ θ log n (θ > 0); m − 1 r = , log n µ− s∗(n, m) (log n)m−1 1 1 g(k)(r) = + p (m) + Zµ(m, n) ,(281) ( − ) ( + ) ( )k k n! n m 1 ! 1 r k=2 log n ∗ where pk(m) is a polynomial in m of degree [k/2] (k = 0, 1, 2,...,m −1), g(w) = 1 , µ is any ﬁxed positive integer such that 2 ≤ µ ≤ m, and (1 + w) ( )m Kµ µ/2 log n Zµ(m, n) = O (m − 1) + (log n)µ m!n

518 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS with (µ) |g (z)|≤Kµ for |z|≤θ. In particular, m − 1 m − 1 (m − 1)(m − 3) p (m) =− , p (m) = , p (m) = , 2 2 3 3 4 8 (m − 1)(5m − 11) (m − 1)(3m2 − 32m + 53) p (m) =− , p (m) =− 5 30 6 144 In general,

[k/2]∗ ∗∗ c2 c3 ...( − )ck k− j 1 2 k 1 pk(m) = (−1) (m − 1) j , ... c2 c3 ... ck j=1 c2!c3! ck!2! 3! k! where ∗∗ indicates that 2c2 + 3c3 + ···+kck = k, c2 + c3 + ···+ck = j, and c2, c3,...,ck ≥ 0. The following recurrence holds true:

~~(k + 1)pk+1(m) =−kpk(m) + (1 − m)pk−1(m)(k ≥ 1),~~

p0(m) = 1, p1(m) = 0 Result (281) is derived via applications of the singularity analysis of P. Flajolet and A. M. Odlyzko [160], and A. Selberg’s method [397]. According to a remark in [397] related to the distribution of the arithmetic function ω(n) (number of distinct prime factors of n, see D. S. Mitrinovic´ and J. Sandor´ [312], Chapter 5), we could replace the principal term of (281) by (log n)m/(m!n),but then the error term would become worse when m = θ(log n). R. Chelluri, L. B. Richmond and N. M. Temme [104] have deﬁned s∗(x, y) for positive real x and y by 1 (u + x + 1) du s∗(x, y) = · ,(282) y+1 2πi C (u + 1) u where C is any Jordan curve which circles the origin in the counterclockwise sense. (u + x + 1) Note that is a polynomial in u for integral x and has no singularities (u + 1) for positive nonintegral x for |u| < x. Let ψ(u) = log[(u + 1)(u + 2)...(u + [x])] − y log u; u1 be the unique positive solution of ψ (u) = 0 (see [439] for a proof that y u is unique), and let t = and B = ψ(u ) − x log(1 + t ) + y log t . Finally, 1 1 x − y 1 1 1 − 1 [x] 1 put g(t ) = [y(x − y)/xψ (u )]1/2 and let H = y − u2/(u + h)2. Then 1 u 1 1 1 1 h=0 x s∗(x, y) ∼ eB g(t ) uniformly for 0 < y ≤ x as x →∞,when x and y are 1 y

519 CHAPTER 5 integers (this is exactly (276)). Moreover if 0 < y ≤ (log x)1/2 then for integral x and y x!(log x + γ)y−1 s∗(x, y) = {1 + O[(log x)−1/2]} (283) y! If (log x)1/2 < y ≤ x − x1/3 then for real x and y ( + + ) ∗( , ) = x 1 u1 + ( / ) ( ) s x y + [1 O 1 y ] 284 ( )( π )1/2 y 1 u1 2 H u1 where the O-constant is independent of y.Ifx − x1/3 ≤ y ≤ x, then independently of y for integral x and y, x + 1 y x−y s∗(x, y) = [1 + O(x1/3)] (285) y + 1 2 The proofs are based on the Moser-Wyman [318] method, as well as that of Temme [439]. For Stirling numbers of fractional order s(a, k) (a ∈ R, k ∈ N), see (90), (92) or (97), (102)-(103). F. Asymptotic expansions of Stirling numbers of the second kind S(n, m) can be found in several papers, too. The asymptotic behavior of S(n, m) for n and m large seems to have been obtained ﬁrst by Laplace (see e.g. L. Moser and M. Wyman [319], F. N. David and E. E. Barton [128]). In recent times, Ch. Jordan [237] (p. 173) has given the formulae

mn n2m S(n, m) ∼ , S(n, n − m) ∼ (n →∞)(286) m! m!2n These formulae, however are applicable only for small values of m.In1948 L. C. Hsu [225] obtained the complete asymptotic expansion n2k f (k) f (k) f (k) S(n + k, n) = 1 + 1 + 2 + ···+ t + O(n−t−1) (t < k), 2kk! n n2 nk (287) where f1, f2,... are polynomials in k whose coefficients can be found; e.g. 1 1 f (k) = (2k2 + k), f (k) = (4k4 − k2 − 3k), 1 3 2 18 1 f (k) = (40k6 − 60k5 − 2k4 − 63k3 + 133k2 − 48k). 3 810 This is a generalization of (286), but it is shown to be a complete asymptotic expansion for fixed k (i.e. when m is fixed in (286)).

~~520 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~Moser and Wyman [319] have given the exact formula~~

− n m1 ( , − ) = −m m k ( ) S n n m q Ak q 288 m k=0 2 dm = m = ( (w) w) (w) w where q , Ak Pk e , with Pk polynomials in ob- n − m dwm w=0 ∞ k tained by the MacLaurin expansion of F(q,w) = pk(w)q about q = 0for k=0 ∞ F(q,w)= exp (−1)k+12ζ(2k)w2kq2k−1 /k(2π)2k. k=1 2 4 Forexample, P0(w) = 1, P1(w) = w /12, P2(w) = w /288, P3(w) = w6/10368 − w4/1440. One ﬁnds that n + k n k (k) (k) (k) (k) 1 S(n + k, n) = 1 + 2 + 4 + 6 − 4 + ... , n 2 6n 72n2 1296 180 n3 (289) which has a similar form to (287). However, this is an exact formula. Moser and Wyman√ [319] proved that (289) behaves as an asymptotic formula for large n when k = o( n). For the range 0 < m < n, lim (n − m) =∞, the following is proved in [319]: n→∞ ∞ ∞ 2 k S(n, m) ∼ Q exp(−t )b2kdt /(mR) ,(290) −∞ k=0 where R is given as the solution of R(1 − e−R)−1 = n/m, Q = A/(mRH)1/2, with R R R 2 m n H = e (e −1− R)/2(e −1) , A = n!(exp(R)−1) /2π R m!, b0 = 1, and b2k are polynomials in φ containing only even powers of φ, with φ = θ(mRH)1/2, |θ | <π. Forexample, if we compute the ﬁrst two terms in (290), we obtain the following expression: n!(exp(R) − 1)m 1 15(C (R))2 3C (R) S(n, m) ∼ 1 − 3 − 4 (291) 2Rnm!(πmRH)1/2 mR 16R2 H 3 4RH2 where 1 C (R)= [R+(R−3R2 + R3)(eR −1)−1 +(3R3 −3R2)(eR −2)−2 +2R3(eR −1)−3], 3 6 1 C (R) = [R + (R − 7R2 + 6R3 − R4)(eR − 1)−1+ 4 24 +(18R3 − 7R2 − 7R4)(eR − 1)−2 + (12R3 − 12R4)(eR − 1)−3 − 6R4(eR − 1)−4].

~~521 CHAPTER 5~~

W. E. Bleick and P. C. C. Wang [51] by using the method of Moser and Wyman have obtained a complete asymptotic expansion of S(n, m) in powers of (n + 1)−1. Convergence is demonstrated for m <(n + 1)2/3/[π + (n + 1)−1/3]. The expansion when divergent is still useful when used as an aymptotic series. For example, the ﬁrst two terms give: n!(eu − 1)K 2 + 18G − 20G2(e4 + 1) S(n, m) ∼ 1 − + (2π(n + 1))1/2m!un(1 − G)1/2 24(n + 1)(1 − G)3 3G3(e2u + 4eu + 1) + 2G4(e2u − eu + 1) + (292) 24(n + 1)(1 − G)3 where ∞ n + 1 u = t − mm−1(te−t )m/m!, t = , m=1 m ∞ u 1 K = (n + 1)(1 − e−u)/u, G = = 1 − u + B u2k/(2k)! u − 2k e 1 2 k=1 The bracket expression in (292), augmented by an additional inverse power of (n + 1),isapproximated by

1 1 1 − + for small u, 6u(n + 1) 72u2(n + 1)2 and by 1 1 1 − + for large u. 12(n + 1) 288(n + 1)2 By using the ray method, Ch. Knessl and J. B. Keller [255] have obtained the following asymptotic expansions for n 1:

~~− n2 n m 1 S(n, m) ∼ if n − m = O(1); (293) 2 (n − m)!~~

~~− / 1 1 2 ∼ 2π m − 1 − n · exp[m − n + n log(mb) − m log(bm − m)], b if n/m ∈ (1, ∞) where b is deﬁned by n b = (b − 1) exp , b ∈ (1, ∞) bk~~

~~522 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

√ D. Gardy [174] has shown that if e.g. n − m = o( m) and n − m →∞, then n m m−k S(n, m) = (1 + o(1)); (294) m 2 and if n − m →∞, with n − m = o(m2/3), then − n m m k 2 S(n, m) = e−(n−m) /6m(1 + o(1)) (295) m 2 As we can see, (294) is similar to (274) for s∗(n, m),but (295) differs from the similar formula (275) for s∗(n, m). N. M. Temme [439] proved the relation n S(n, m) ∼ eAmn−m f (t ) (296) 0 m n − m uniformly as n →∞for δ

m − x = 1 − e x0 and A = φ(x ) − mt + (n − m) log t , n 0 0 0 0 where φ(u) =−n log u + m log(eu − 1). In 1999 P. Flajolet and H. Prodinger [161] have deﬁned the Stirling numbers of complex arguments x, y by x! 1 S(x, y) = · z−x−1(ez − 1)ydz,(297) y! 2πi C where C is a Hankel contour which starts at −∞, circles the origin and goes back to −∞ subject to |Im z| < 2π. Here x! = (x + 1), where is Euler’s gamma function. R. Chelluri, L. B. Richmond and N. M. Temme [104] have generalized relation (296) to real numbers x and y,byusing the deﬁnition (297). Let u0 be the unique y − solution of u = 1 − e u0 .Let x 0 x − y t = ,φ(u) =−x log u + y log(eu − 1), 0 y

~~A = φ(u0) − yt0 + (x − y) log t0, 1/2 t0 u u −1 u u −2 f (t0) = , 2H0(u) = e (e − 1) − ue (e − 1) . (1 + t0)(u0 − t0)~~

~~523 CHAPTER 5~~

~~Then x S(x, y) ∼ eA yx−y f (t ) (298) 0 y uniformly as x →∞for δ0isﬁxed). Furthermore if δ~~

~~524 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

Elbert proves that Fn converges weakly to a distribution function F, and using Stieltjes transforms determines certain representations for F and F . Previous results for Pn(1) are well known, see F. E. Binet and G. Szekeres [49] and N. de Bruijn [65]. For the asymptotic approximations of r-Stirling numbers, see R. B. Corcino, L. C. Hsu and E. L. Tan [121]. An asymptotic formula for the Bell numbers was given by Moser and Wyman (see relation (262)). For an asymptotic relation for a generalization of Bell numbers, see C. B. Corcino [119]. For asymptotics on Stirling numbers for complex argument, see G. Kemkes, C. F. Lee, D. Merlini and B. Richmond [247]. For another kind of generalization of s∗(n, k) to s∗(z, k) where z ∈ C, k ∈ N,involving hypergeometric functions, see V. Adamchik [2].

~~5.3 Bernoulli and Euler numbers~~

1 Deﬁnitions, basic properties of Bernoulli numbers and polynomials As we have seen in 5.2., the Bernoulli numbers (see (14) of 5.2.) are strongly related to Stirling numbers, too (see e.g. relation (15) of 5.2.). For general references, see [334], [373], [144], [142]. For Bernoulli numbers of higher order, connected to Stirling numbers, see (63), (210) of 5.2. A. Recall that the coefﬁcient Bm in the expansion ∞ t B = n tn (1) t − e 1 n=0 n! is called the nth Bernoulli number. We note that there is another (older) notation for Bernoulli numbers (see e.g. T.

J. I’A. Bromwich [64], p. 297) when the Bernoulli numbers Bn are defined by ∞ t t (−1)n−1 B = 1 − + n t2n (1 ) t − ( ) e 1 2 n=1 2n ! ( ) ( ) It is easy to see that the coefficients Bn are connected with Bn given by(1) via =| |=(− )n+1 ( ≥ ) Bn B2n 1 B2n n 1 Ramanujan (see Berndt [46], I.) defined the signless Bernoulli numbers of arbitrary index s by writing ∗ = ( + )(ζ( )( π)−s Bs 2 s 1 s 2

~~525 CHAPTER 5~~

∗ = (− )m+1 > ( ≥ ) By (7) and (8), clearly B2m 1 B2m 0 m 1 . The Bernoulli numbers ﬁrst appeared in Jacob Bernoulli’s (1654-1705) posthu- mous work ”Ars Conjectandi” (1713) in connection with sums of powers of the form

~~k k k fk(n) = 1 + 2 + ···+n (k ≥ 0 integer)~~

The numbers fk(n) can be calculated via the recurrence relation k + 1 k + 1 (n + 1)k+1 − 1 = f (n) + f (n) + ···+ f (n), (2) 0 1 1 k k where f0(n) = n.But they can be explicitly obtained by the formula k+1 + 1 k 1 j f (n − ) = B + − n ( ) k 1 + k 1 j 3 k 1 j=1 j

2 For identities of type ( f1(n)) = f3(n), and related divisibility problems, see D. E. Penney and C. Pomerance [345]. From the deﬁnition (1) follows at once that n n + 1 B0 = 1, Bk = 0 for k ≥ 1; k=0 k and that B2m+1 = 0form ≥ 1 (4) Since B x x x ex + 1 x x n xn − B x = + = · = coth , 1 x − x − n≥0 n! e 1 2 2 e 1 2 2 we can easily deduce the expansions 1 B cothx = 2n · 4n x2n, x (2n)! n≥0 ( ) B π 5 x = 2n n( n − )x2n−1 |x| < tanh ( ) 4 4 1 n≥1 2n ! 2 The similar formulae for tg x and cotx are: B x = (− )n−1 2n n( n − )x2n−1, tan 1 ( ) 4 4 1 n≥1 2n ! (6) 1 B x = (− )n 2n n x2n cot 1 ( ) 4 x n≥0 2n !

~~526 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~We note that similar expansions are the following (see [64])~~

~~∞ x 2(22n − 1) = + (− )n−1 B x2n (|x| <π) 1 ( ) 1 2n sin x n=1 2n !~~

∞ sin x 22n−1 = (− )n B x2n (|x| <π), log ( ) 1 2n x n=1 n 2n ! ∞ 22n−1(22n − 1) π x = (− )n−1 B x2n |x| < log cos ( ) 1 2n n=1 n 2n ! 2 ∞ cosh x − cos x nπ (−1)n B x2n π log = 2n+1 cos · 2n |x| < 2 ( ) x n=1 2 2n 2n ! 2 which follow by integration of known power series expansions (e.g. of (6)). One of the most important properties is given by Euler’s formula 2(2k)! B = (−1)k+1 ζ(2k)(7) 2k (2π)2k ∞ where ζ(s) = n−s (Re s > 1) is the Riemann zeta function. A direct consequence n=1 of (7) is k+1 (−1) B2k > 0 for k ≥ 1 (8) ∞ zn By deﬁning the tangent numbers T (n) by tan z = T (n) , T (0) = 1, we n=0 n! immediately get from (6) and (8) that

22n−1(22n − 1) T (2n) = 0, T (2n − 1) = |B(2n)| (9) n As an application of (9) and the von Staudt-Clausen divisibility theorem for B(2n) (which we shall study later), R. Mcintosh [232] proved the following primality condition for a Fermat prime

~~n F(n) = 22 + 1 (n ≥ 0) :~~

~~F(n) is prime if and only if F(n) T (F(n) − 2)(10)~~

~~B. The Euler numbers (En) are deﬁned by ∞ 2 E = n tn (E = 1, E =−1, E = 5, E =−61,...) (11) t + −t 0 2 4 6 e e n=0 n!~~

~~527 CHAPTER 5~~

~~In fact, Euler considered the series expansion of the secant function~~

∞ 1 (−1)n E π = x = 2n x2n |x| < ( ) sec ( ) 12 cos x n=0 2n ! 2 these numbers were later called Euler numbers by H. F. Scherk (1825). We note that there is in use another notation for the Euler numbers, namely (see e.g. [64]) ∞ E π x = n x2n |x| < ( ) sec ( ) 12 n=0 2n ! 2

~~Then clearly, these coefﬁcient are connected with (En) by =| | ( ≥ ) En E2n n 0~~

~~There is also a third notation, namely (see e.g. [238])~~

~~∞ n = ∗ x ( ) sec x En 12 n=0 n!~~

~~Then by (12) and (12 ) these coefﬁcient satisfy~~

~~∗ = = (− )n , ∗ = ( ≥ ) E2n En 1 E2n E2n+1 0 n 0 1 The expansion of cosecx = is given by sin x ∞ 1 (−1)n+1 B (22n − 2) x = + 2n x2n−1 (|x| <π, x = )( ) cosec ( ) 0 12 x n=1 2n !~~

~~For the numbers (S2n) given by the expansion~~

~~∞ S x x = 2n x2n sec ch ( ) n=0 2n ! see L. Carlitz [80], Gandhi and Singh [173]. The Euler numbers satisfy n 2n E2k = 0, E2k+1 = 0 (k = 0, 1,...) (13) k=0 2k~~

~~From (4) it follows that the Bernoulli numbers are rational numbers, while (9) and~~

~~528 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

(13) imply that the tangent and Euler numbers are always integers. For the computation of tangent, Euler and Bernoulli numbers, see T. Buckholtz and D. Knuth [66]. C. The Bernoulli and Euler polynomials Bn(x), En(x) are deﬁned by

∞ ∞ text tn 2ext tn = B (x) (|t| < 2π); = E (x) (|t| <π) (14) t − n t + n e 1 n=0 n! e 1 n=0 n! Then it is immediate that 1 B = B (0) and E = 2n E , B (1) = B (n = 1), B (1) =−B (15) n n n n 2 n n 1 1

~~Therefore, En = En(0);infact~~

2 n+1 E (0) =−E (1) = (1 − 2 )B + ,(16) n n n + 1 n 1 but 1 1 B =− 1 − B (17) n 2 2n−1 n Forvalues at other rational points, we note that 1 2 1 1 B = (−1)n B =− 1 − B (n = even); n 3 n 3 2 3n−1 n 1 n 2 2 1 n+1 1 E = (−1) E = B + − 2 B + = (18) n 3 n 3 n + 1 n 1 3 n 1 6 1 n+1 1 = (1 − 2 ) 1 − B + (n = odd); n + 1 3n n 1   − 1 − 1 ( = )  1 Bn n even 2n 2n−1 1 n 3 Bn = (−1) Bn = (19) 4 4   n − E − (n = odd) 4n n 1 1 5 1 1 1 B = (−1)n B = 1 − 1 − B (n = even)(20) n 6 n 6 2 2n−1 3n−1 n In (15)-(20) there are listed all the known values of Bernoulli and Euler polynomials at rational points, expressed in terms of one and only one Bernoulli or Euler number. It is unknown whether there exist corresponding such formulas for values at other rational points.

~~529 CHAPTER 5~~

~~See A. Granville and Z. W. Sun [201], where connections to Fermat quotients are also pointed out. If m ≥ 3, and n ≥ 2iseven, then one has (see [201])~~

~~1 B (k/m) = (1 − pn−1)B (21) n n−1 n 1≤k~~

~~The basic properties of these polynomials are the following:~~

~~m m m B + (m + 1) − B + E (m + 1) + (−1) E (0) kn = n 1 n 1 , (− )m−kkn = n n + 1 k=1 n 1 k=1 2 (22) for m, n = 1, 2,...~~

~~B (1 − x) = (−1)n B (x), E (1 − x) = (−1)n E (x) n n n n ( ) n n−1 n+1 n 23 (−1) Bn(−x) = Bn(x) + nx ,(−1) En(−x) = En(x) − 2x for n = 0, 1, 2,...~~

B (x) = nB − (x), E (x) = nE − (x)(n = 1, 2,...) n n 1 n n 1 ( ) n−1 n 24 Bn(x + 1) = Bn(x) + nx , En(x + 1) =−En(x) + 2x (n = 0, 1,...) x Bn+1(x) − Bn+1(a) Bn(t)dt = (n ≥ 1) a n + 1 x En+1(x) − En+1(a) En(t)dt = (n ≥ 0) a n + 1 1 n−1 m!n! Bn(t)Bm(t)dt = (−1) Bm+n (m, n ≥ 1) 0 (m + n)! ( ) 25 1 n m+n+2 m!n! En(t)Em(t) = (−1) · 4(2 − 1) Bm+n+2 (m, n ≥ 0) 0 (m + n + 2)! n n n n−k n n−k Bn(x + h) = Bk(x)h , En(x + h) = Ek(x)h (26) k=0 k k=0 k m−1 n−1 k Bn(mx) = m Bn x + (n ≥ 0, m ≥ 1) (Raabe’s theorem) (27) k=0 m m−1 n k k En(mx) = m (−1) En x + (n ≥ 0, m ≥ 1 odd) k=0 m

~~530 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~m−1 2 n k k E (mx) =− m (− ) B + x + (n ≥ , m ≥ ), ( ) n + 1 n 1 0 2even 28 n 1 k=0 m~~

− n n E 1 n k E (x) = k x − (n ≥ )() n k 0 29 k=0 k 2 2 There are true also the following Fourier expansions: 1 ∞ cos 2πkx − πn n! 2 B (x) =−2 n ( π)n n 2 k=1 k (n > 1, 0 ≤ x ≤ 1; n = 1, 0 < x < 1) (30) 1 ∞ sin (2k + 1)πx − πn n! 2 E (x) = 4 n π n+1 ( + )n+1 k=0 2k 1 (n > 1, 0 ≤ x ≤ 1, n = 0, 0 < x < 1)

∞ (−1)n−1 · 2(2n)! cos 2kπx B (x) = 2n ( π)2n 2n 2 k=1 k (n ≥ 1, 0 ≤ x ≤ 1) (31) ∞ (−1)n · 4(2n)! sin(2k + 1)πx E (x) = 2n π 2n+1 ( + )2n+1 k=0 2k 1 (n ≥ 1, 0 ≤ x ≤ 1, n = 0, 0 < x < 1)

∞ (−1)n · 2(2n − 1)! sin 2kπx B − (x) = 2n 1 ( π)2n−1 2n−1 2 k=1 k (n > 1, 0 ≤ x ≤ 1, n = 1, 0 < x < 1) (32) ∞ (−1)n · 4(2n − 1)! cos(2k + 1)πx E − (x) = 2n 1 π 2n ( + )2n k=0 2k 1 (n ≥ 1, 0 ≤ x ≤ 1) There exists the following relation between the two polynomials: n + 2 x 1 x 2 n x E − (x) = B − B = B (x) − 2 B (n ≥ 1)(33) n 1 n n 2 n 2 n n n 2

~~531 CHAPTER 5~~

For these, and other fundamental properties, see [237], [338], [1], where also tables can be found. For polynomials related to Bernoulli numbers of higher order, see L. Berg [44]. For the zeros of Bernoulli, as well as generalized Bernoulli and Euler polynomials, see K. Dilcher [140] and the references therein. D. The Genocchi numbers Gn are deﬁned by (see e.g. [116])

~~∞ 2t G = n tn (|t| <π) (34) t + e 1 n=0 n!~~

~~It follows that G0 = 0, G1 = 1 and G2n+1 = 0 (n ≥ 1). The numbers G2n are connected to the Bernoulli numbers by~~

~~2m G2n = 2(1 − 2 )B2n,(35) while to E2n−1(0) by G2n = 2nE2n−1(0)(36) The Genocchi polynomials are deﬁned by~~

~~∞ tn 2t G (x) = etx (37) n t + n=0 n! e 1~~

~~It follows that~~

~~n−1 Gn(x + 1) + Gn(x) = 2nx (n ≥ 1); Gn =−Gn(1)(n > 1);~~

~~n−1 n−1 Gn(x) + (−1) Gn(−x) = 2nx ;~~

~~∗ [N/2] n n n−2i Gn(x) + (−1) Gn(−x) = 2n G2i x , where N = n (38) i=1 2i or n − 1, according as n is even or odd;~~

~~n−1 Gn(x) + Gn(x − 1) = 2n(x − 1) ;~~

~~Gn(1 − x) = Gn(x) if n is odd; Gn(1 − x) =−Gn(x), if n is even.~~

~~Gn(x) is connected with Bn(x) by x + 1 x G (x) = 2n B − B (39) n n 2 n 2~~

~~532 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

One has the following multiplication theorems m−1 n−1 k k Gn(mx) = m (−1) Gn x + , if m is odd; k=0 m m−1 n−1 k k Gn(mx) =−2m (−1) Bn x + , if m is even (40) k=0 m Forspecial values, we note that, for n even (n > 1) 1 1 3 1 2 G = 0, G (1) =−G , G = G , G =−G ; n 2 n n n 4 n 4 n 3 n 3 while for n odd, 1 3 1 2 G (1) = G = 0, G = G , G = G (41) n n n 4 n 4 n 3 n 3

~~Generally, n 2n 2m 1 G + = , + 2m 1 0 m=0 2m 2m 1 2 ( ) 42 22m − 1 1 G = 24m B 2m 22m−1 − 1 2m 4~~

~~Relationships between Gn and En are e.g. n n−1 − 1 n j 2n j n 1 E − = G , G = (− ) E − − ( ) n 1 2 j n n 1 n 1 j 43 2n j=0 j 2 j=0 j~~

(k)( ) Genocchi polynomials of order k, Gn x are deﬁned by (see A. F. Horadam [218]) ∞ tn 2t k G(k)(x) = etx (k = 0, 1,...) (44) n t + n=0 n! e 1 whence (k)( ) = < , (0)( ) = n, Gn x 0forn k and Gn x x (k) = (k)( ) having similar properties. The Genocchi numbers of order k are Gn Gn 0 . For logarithmic polynomials and numbers,see J. M. Gandhi [172], [170].

~~533 CHAPTER 5~~

2 Identities In what follows we shall select some identities and recurrence relations involving Bernoulli and Euler numbers. 1 A. By computing in two different ways the integrals (x + 1)Bk(x + 1)dx, and 0 1 xBk(x)dx, one can deduce (see J. Sandor´ [386]): 0 k k 2k−i+2 − 1 1 B = (B + + k + ) ( ) i − + k 1 2 1 + 45 i=0 i k i 2 k 1 By putting k = 2m this implies m 2m 22m−2i+2 − 1 4m + 1 m B = + ( 2m+1 − )(m ≥ )( ) 2i − + + + 2 1 1 45 i=0 2i m i 1 2m 1 2m 1 Similar identity is m 2m 1 2(m − 1) B = (m ≥ )() 2i − + + 1 46 i=0 2i m i 1 2m 1 For the identity m−1 − 2m 1 2i B2i · 2 = 2m − 1 (m ≥ 2)(47) i=0 2i see also [386], where a new proof od k−1 ( 2i − ) − ( 2k − ) 2 1 B2i 2k 1 = 1 − 2 1 B2k ( ) − 2 48 i=1 2i 2k 2i 2 2k is given too. For a proof of (48) by contour-integration arguments, see Gh. Stoica [425]. The following formula has been proved by Gelfand (see [144], [142]): k m k B + m B + k!m! (− )m−1 m i + (− )k−1 k j = ( ) 1 − + 1 − + ( + + ) 49 i=1 i 1 m i j=1 j 1 k j k m 1 ! By setting m = k in (49), we have k (− )k−1 ( )2 k Bk+i = 1 · k! ( ) − + ( + ) 49 i=1 i 1 k i 2 2k 1 !

~~534 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~B. The following identities can be found in T. Agoh [7]. k+r + + + k r m r i k+m+r−1 Bm+i + (−1) · i=0 i r~~

+ m r m + r k + r + j · Bk+ j = 0; (50) j=0 j r k+r + + + m k r m r i k+r−1−i (−1) 2 Em+i + i=0 i r m+r + + + k m r k r j m+r−1− j +(−1) 2 Ek+ j = j=0 j r r k + r m + r = (− )s ( ) 1 − 51 s=0 s r s By letting k = m in (50) and then in (51), one can deduce k+r k + r k + r + i Bk+i = 0 (r ≥ 1, odd), (50 ) i=0 i r k+r k + r k + r + i + − + r k + r k r i = (− )k 2 ( ≥ , )( ) 2 Ek+i 1 r r 0 even 51 i=0 i r 2 If r = 1in(50 ) and replace k by k − 1, and then letting r = 0in(51 ) one obtains the following classical formulas which were repeatedly discovered by many authors: k k (k + i)Bk−1+i = 0 (50 ) i=0 i k k k−i k 2 Ek+i = (−1) (51 ) i=0 i We quote also the following formulas from [7]: k k B + + ( m+1+i − ) m 1 i + 2 1 + + i=0 i m 1 i m m B + + +(− )k+m−1 ( k+1+ j − ) k 1 j = ,() 1 2 1 + + 0 52 j=0 j k 1 j

~~535 CHAPTER 5~~

∗ [(k+1)/2] + k 1 2k+2−2i (2 − 1)B2k+2−2i = 0,(53) i=0 2i ∗ [(k+1)/2] + k 1 2k+2−2i 2i k−1 k (2 − 1)3 B2k+2−2i = (−1) 2 (k + 1)(54) i=0 2i C. The following famous identity is due to Euler (see B. C. Berndt [46], I): If n ≥ 2, then

~~n (2n)! B B − =−( n + )B ( ) ( ) ( − ) 2k 2n 2k 2 1 2n 55 k=1 2k ! 2n 2k !~~

J.-I. Igusa [231] obtained a similar identity by the use of Eisenstein series: For each positive integer n ≥ 4 one has n−2 ( − ) ( + )( − ) 2n 2 ! · B2k · B2n−2k = − B2n 2n 1 2n 6 ( ) ( − ) ( − − ) − ( − )( − ) 56 k=2 2k 2 ! 2n 2k 2 ! 2k 2n 2k 2n 6 2n 2 2n 3

Fornew proofs of (55) and (56) via the Shintani zeta functions, see M. Eie and K. F. Lai [149]. They prove also the following new Bernoulli identity: For any integer m ≥ 3, (2m − 4)! − − − − · p1 1 · p2 1 · p3 1 · p4 1 = Bp1 Bp2 Bp3 Bp4 2 3 5 6 |p|=2m p1!p2!p3!p4! 1 B 17 1 257 B − =− · 2m − + · 22m−2 + · 32m−4 2m 2 − 1080 2m 432 48 360 2m − 2 197 62m−4 1 1 − · B − + B − + 180 2m − 2 2m 2 6 2m 2 3 1 62m−4 1 1 52m−3 1 + · 25B − − 16B − − B − ,(57) 54 2m − 3 2m 3 6 2m 3 3 2m − 3 2m 3 5 where |p|=p1 + p2 + p3 + p4 and p = (p1, p2, p3, p4), with p1 ≥ 0 integers (i = 1, 4). We quote also the following famous Ramanujan identities (see [46]), Part I, p. 122): Let α, β > 0 with αβ = π 2. If n ≥ 2, then

~~∞ ∞ k2n−1 k2n−1 B αn − (−β)n = (αn − (−β)n) 2n ; (58) 2αk − 2βk − k=1 e 1 k=1 e 1 4n~~

~~536 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

n+1 B B + − 2k · 2n 2 2k ( k − )( n − k + )(−β)kαn+1−k = ( ) ( + − ) 2 1 2 2 1 k=0 2k ! 2n 2 2k ! ∞ ∞ csch2kα csch2kαcothkα = (2n − 1)2−1−2nα1−n + 2−2nα2−n + 2n 2n−1 k=1 k k=1 k ∞ ∞ csch2kβ csch2kβcothkβ +(2n − 1)2−1−2n(−β)1−n − 2−2nβ2−n (59) 2n 2n−1 k=1 k k=1 k D. Identities between Bernoulli numbers are used many times in the proof of some congruences. In [219] F. T. Howard provides an easy proof of the recurrence

− − 1 m1 m n 1 B = nk B j m−k,(60) m ( − m) k n 1 n k=0 k j=1 and uses this formula to the proof of von Staudt-Clausen, Frobenius, etc. -type theorems. M. Kaneko [241] puts Bn = (n + 1)Bn, and gives two proofs of the identity n−1 1 n + 1 B =− B + ,() 2n + n k 61 n 1 k=0 k ( ) = one√ of which√ uses convergents of continued fraction expansions of f x ( x/2)coth( x/2), and a very short one (due to D. Zagier) deﬁning a suitable convolution on the set of sequences. For double series we note that in 1970 J. Higgins [217] proved n m m kn B = (− )k ( ) n 1 + 62 m=0 k=0 k m 1

~~S. Bhargava and C. Agida [47] have shown that~~

~~− n n 1 m (−1)m+k m B = (m − k)n−1 (63) n − n m+1 1 2 m=0 k=0 2 k~~

~~As a generalization of (63), H. Alzer obtained ([22]):~~

~~− n 1 m (−1)m+k m B (2x) = 2n B (x) + n (m − k + 2x)n−1 (64) n n m+1 m=0 k=0 2 k~~

~~537 CHAPTER 5~~

~~For x = 0 this implies (63). In 1989 K. Dilcher [141] extended (64) to n−1 m n k+1 m n−1 Bn(λx) = λ Bn(x) + n (−1) Aλ(m)(k + λx) ,(65) m=0 k=0 k~~

λ−1 i2π j/λ i2π j/λ m+1 where Aλ(m) = (e )/(1 − e ) (λ ≥ 2isaninteger). j=1 Further, n Bn(λx) = λ Bn(x)+ ∗ [(n−1)/2] 2m+1 (− )m+k + 1 2m 1 n−1 +n Cλ(k, m)(m − k + λx) ,() + 66 m=0 k=0 2m 1 k where (λ−1)/2 −2m−1 2m+2 Cλ(k, m) =−2 {k + (2m + 1 − k) cos 2 jπ/λ}{sin( jπ/λ) } j=1 and λ ≥ 3isanodd integer. By making use of the Alzer and Dilcher results, P. G. Todorov [447] gave twelve explicit formulas for the Bernoulli and Euler polynomials and numbers, e.g. n m−1 n n k+1 n n−1 Bn = (n/2 (2 − 1)) (−1) k (n ≥ 1)(67) m=1 k=0 m

M. Hauss [214] has introduced the ”conjugate” Bernoulli polynomials Bn(x) by applying the Hilbert transform to the (1-periodic) Bernoulli polynomials Bn(x) = H1 Bn(x), x ∈ [0, 1).The conjugate Bernoulli numbers are deﬁned by Bn = Bn(0). The following remarkable formula for ζ(2m + 1) (m ≥ 1 integer) is derived:

B + ζ(2m + 1) = (−1)m22mπ 2m+1 2m 1 (m ≥ 1)(68) (2m + 1)! E. T. Agoh, K. Dilcher and L. Skula [12] have studied the Euler quotients aϕ(m) − 1 q(a, m) = , where m ≥ 2 and a are two relatively prime integers. By m Euler’s divisibility theorem (see Chapter 3), q(a, m) is an integer. The Euler quotients (called also as generalized Fermat quotients by M. Lerch [282]) are connected to Bernoulli polynomials and numbers by the identity (see [12]):

~~− aϕ(m) a 1 j aq(a, m) =− Bϕ(m) − Bϕ(m) (69) ϕ( ) mB m j=1 a~~

~~538 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

3 Congruences for Bernoulli numbers and polynomials. Eulerian numbers and polynomials The congruences and divisibility properties of Bernoulli and Euler numbers and polynomials are of fundamental importance in many fields of mathematics (for example in the study of the cyclotomic field, see e.g. K. Iwasawa [234], P. Ribenboim [363]). 1 1 1 The first few nonzero terms of (B ) are B = 1, B =− , B = , B =− , n 0 1 2 2 6 4 30 1 1 5 691 7 3617 B = , B =− , B = , B =− , B = , B =− , 6 42 8 30 10 66 12 2730 14 6 16 510 43867 B = . B is the first one with a composite numerator 174611 = 283 · 617. 18 798 20 Nn A. Write Bn = with Dn > 0 and (Nn, Dn) = 1. The denominators are Dn completely characterized by the famous von Staudt-Clausen theorem from 1840 (see [422], [115]): If n ≥ 2, even, then Dn = p,(70) pprime,(p−1)|n and Bn + 1/p is an integer.(71) pprime,(p−1)|n

If Zp denotes the ring of p-adic integers, we see that as a consequence of the above theorem, if (p − 1) n, the Bn ∈ Zp. Also if (p − 1)|n, then pBn ∈ Zp (more precisely, pBn ≡−1 (mod p)). (The relation Bn ∈ Zp may be expressed also by saying that Bn is p-integral). More generally, if a, b, c, d, m (m > 1) are integers. and if (b, m) = 1, we a a c a c say that is m-integral. If and are m-integral, we deﬁne ≡ (mod m) iff b b d b d m|(ad − bc), i.e. if ad ≡ bc (mod m) with the usual deﬁnition of congruence. It is easily seen that the familiar properties of congruence on Z continue to hold. In 1845 von Staudt [421] proved a related result, too: If n ≥ 2iseven, and p is a r r prime with (p − 1) n, then if p |n for some r ≥ 1, then p |Nn, too. (72) There are many proofs known in the literature for these theorems. For some recent proofs, see Eie and Lai [149] (where (70)-(71) are proved via certain identity for Bernoulli numbers), K. Girstmair [190] (where (72) is proved with the introduction k t of certain ”cyclotomic” Bernoulli numbers Bm,k (0 ≤ k ≤ n−1),satisfying t/(ζ e − ∞ m 1) = Bm,kt /m!, where ζ is a primitive nth root of unity for n ≥ 2), A. Robert m=0 [369] (who proved (72) by using a p-adic version of the mean value theorem).

~~539 CHAPTER 5~~

There exist many applications of the above theorems. For example, by using the von Staudt-Clausen theorem, R. W. van der Wall [348] (by answering a question ∞ 2r−1 posed by F. Hirzebruch) has shown that if tan x = ar x (|x| <π/2), then r=1 −1 ∈ Z = = ar if and only if r 1orr 2. (73) P. Erdos¨ and S.S. Wagstaff, Jr. [154] have studied the fractional parts of the Bernoulli numbers. By applying among others the von Staudt-Clausen theorem, they proved that the sequence

~~{B2m} is dense in (0, 1)(74)~~

Further, for every k ≥ 1, the set of all m for which {B2m}={B2k} has positive asymptotic density. (75) 1 (Here {x}=x − [x]∗ denotes the fractional part of x). Let F (z) = card{m ≤ x x x : {B2m} < z}.Then the limiting distribution lim Fx (z) = F(z) exists, and is a x→∞ jump function. The convergence is uniform and the sum of the heights of the jumps of F is 1. (76) Forapplications in the theory of Fermat and Euler quotients, e.g. E. Lehmer [275], D. H. Lehmer [276], Agoh-Dilcher-Skula [12]. For example, in [12] it is proved that for p an odd prime, and n a positive integer the following congruence is true: (pn−1)/2 1/k ≡−2q(2, pn) + pnq(2, pn)2 (mod pn+1)(77) k=1,pk This generalized a result by E. Lehmer [275] (see (69) for q(a, m)). For many other congruences on q(a, m), see [12]. For Wieferich primes (i.e. satisfying q(2, p) ≡ 0 (mod p))see also R. E. Crandall, K. Dilcher and C. Pomerance [124]. ForWilson quotient, see [11], [8]. By applications of (70), (71) B. J. Powell [354] proved that if p > 3isaprime such that 2p − 1isalso prime, then

~~|N2p−2| > |N2p|,(78) while in [355] he proposed that for p > 3prime~~

N 2p ≡ 1 (mod p)(79) p and that for each k ≥ 1 there exist inﬁnitely many r, s, t,... such that Nk|Nr , Nk|Ns,... (80) If p is a prime such that 2p + 1iscomposite (e.g. when p ≡ 1 (mod 3)), then N2p is divisible by a prime ≡ 3 (mod 4),see Z. I. Borevic and I. R. Safarevic [57].

~~540 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~The proofs use also the following famous congruence of Kummer [265]. If n ≥ 1andp > 3isaprime such that (p − 1) 2n, then~~

B + − B 2n p 1 ≡ 2n (mod p), (81) 2n + p − 1 2n where the congruence relation is in the sense of Zp. More generally, if m ≡ n (mod (p − 1)pN ) and (p − 1) m, then B B (1 − pm−1) m ≡ (1 − pn−1) n (mod pN+1)(82) m n In 1910 Frobenius [168] gave a generalization of the Kummer congruence. Van- diver [463] obtained the complementarity congruences, which were extended by Car- litz [89] in many directions. In 1964 J. M. Gandhi [171] obtained congruences for Genocchi numbers (see m m m (34), (35)), similar to the above type congruences. Let Zm(n) = 1 − 2 + 3 − ···+(−1)n+1nm. Then, if m ≥ 2, n ≥ 3 odd, F. T. Howard [219] has shown that

2 G2m ≡−2mZ2m−1(n − 1)(mod n ), and that if p ≥ 5 prime, and m > 1, then ( 2m − ) 2 2 1 B2m G2m 2 ≡− ≡ Z − (p − 1)(mod p ) 2m 2m 2m 1 A main ingredient of the proof is the identity m−1 m m k (n − n)Gm = n Gk Zm−k(n − 1) k=1 k (n, m > 1, n = odd). Eie and Lai [149] have shown that (82) follows from a Bernoulli identity, von Staudt-Clausen’s theorem and Euler’s generalization of Fermat’s little theorem. Kummer’s congruence (82) can be restated in p-adic analysis (see N. Koblitz [259]) by saying that the p-adic integrals are continuous on compact sets. Other useful fact is contained in the following theorem of Adams [3]: k If p is a prime, n ≥ 1 and (p − 1) 2n, p |2n for some k ≥ 1, then pk|N2n.(83) I. Sh. Slavutskii [405] (see also [411]) attributes both Kummer’s congruence and Adams’s theorem to two obscure pamphlets [421] of von Staudt. Finally, we mention the following congruence of Voronoi (G. Voronoi (1889)): If p is a prime, p a,and(p − 1) 2n,(a, n positive integers), then

~~− ∗ B p 1 ja (a − a p−2n) 2n ≡ j 2n−1 (mod p), (84) 2n j=1 p~~

~~541 CHAPTER 5~~

The following more general version is true (see e.g. J. V. Uspenski and M. A. Heaslet [460] or K. Ireland and M. Rosen [233]): If a, k ≥ 1, (a, k) = 1, then k−1 ∗ 2n 2n−1 aj (a − 1)N2n ≡ 2nD2n (aj) (mod k)(85) j=1 k

This is essentially due to Voronoi [468]: see also [408] and [353]. B. Various generalizations and extensions, or analogues of the von Staudt- Clausen, Kummer, Adams, and Voronoi-theorems are known in the literature. In what follows we shall study such relations. Let k be a positive integer and p > 3aprime. Z. H. Sun [432] (see also [433]) proved that

~~2 pBk(p−1) ≡ kpBp−1 − (k − 1)(p − 1)(mod p ), and ( ) − 86 k k 1 3 pB ( − ) ≡ pB − − k(k − 2)pB − + (p − 1)(mod p ) k p 1 2 2p 2 p 1 2~~

~~More generally, let k, l, n be positive integers, p a prime, and put h = ϕ(pl ).I. Sh. Slavutskii [411] shows that e.g.~~

k(k − 1) (k − 1)(k − 2) pB ≡ pB − pB k(k −2)+(p−1) (mod p2l+1)(87) kh 2h 2 h 2 for p ≥ l +3. For l = 4 one obtains results of type (86). There are proved also results that generalize theorems due to Vandiver [463], Carlitz [89] and others. We quote also the following congruence from [411]:

~~l l+1 pBhk ≡ p − 1 + kp wp (mod p ) for p ≥ 5,(88)~~

~~(p − 1)! + 1 where w = (integer, by Wilson’s theorem). Relation (88) implies p p~~

l pBhk ≡ p − 1 (mod p ), (89) due to L. Carlitz [89] and other theorems due to Beeger, Lerch, Lehmer (see relation l (108)). In fact Carlitz proved (89) in the form pBhk ≡ p − 1 (mod p ), x ∈ Zp, p ≥ 3, and this is also a particular case of +− m x x p l pB + (x) ≡ pB (x) − p B (mod p ), (90) hk m m m p

~~542 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

for p ≥ 3, x ∈ Zp, l, k ≥ 1 and m ≥ 0, due to Sun [433]. Here −xp is the least nonnegative residue of −x (mod p), and Bn(x) is a Bernoulli polynomial. Theorem (90) is a consequence of a more general result, namely. Let p, l, m, x be as above. Then l +− l k k(p−1)+m x x p (−1) pBk(p−1)+m(x) − p Bk(p−1)+m ≡ k=0 k p ( l ), ∈ ( − ) ≡ 0 mod p if n Sp or p 1 m ( ) l−1 l 91 p (mod p ), if n ∈ Sp and (p − 1)|m, where {3, 5, 7,...}, if p = 2, S ={s : B ∈ Z , m ≥ 1}= p s p {s|p − 1 s, s ≥ 1}, if p > 2

~~Another result by Sun says that +− k(p−1)+m x x p pB ( − )+ (x) − p B ( − )+ ≡ k p 1 m k p 1 m p~~

− l 1 k − 1 − r k ≡ (− )l−1−r · 1 − − r=0 l 1 r r +− r(p−1)+m x x p · pB ( − )+ (x) − p B ( − )+ + r p 1 m r p 1 m p k +(−1)l δ(l, m, p) pl−1 ≡ l l l−1 l ≡ al k + an−1k + ···+ak + a0 (mod p )(k ≥ 0), (92) where a0,...,al are integers, and 1, if B ∈ Z and p − 1|m δ(l, m, p) = l p 0, if Bl ∈ Zp or p − 1 m This is a wide generalization of the von Staudt-Clausen theorem. Using the properties of the Stirling numbers of the second kind, Sun shows also that one can choose s−1 a0,...,al so that ass!/p ∈ Zp for every s ∈{0, 1,...,l}. As example of (92) we note that

2k 2 7 (2 − 2 )B2k ≡ 12(2k) + 18 · 2k + 1 (mod 2 ), 2k 4 3 2 5 (3 − 3 )B2k ≡−36k + 108k − 93k + 18k + 2 (mod 3 ), (93) 4k 4 3 2 4 (5 − 5 )B4k ≡−125k + 125k − 300k − 100k + 4 (mod 5 )

~~543 CHAPTER 5~~

~~G. Frobenius [168] proved that for k > 2,~~

~~2B2k ≡ 1 − 12m (mod 32) and F. T. Howard [219] has extended the Frobenius congruences to~~

~~3 2B2k ≡ 1 − 12k + 16k(1 + k )(mod 64),~~

~~3 2 2B2k ≡ 1 − 12k + 16k(1 + k ) − 32k (1 + k)(mod 128), for k > 3, and more generally l 2k 2 j 2k 2l+1 (1 − 2 )2B2k ≡ 1 − 2k + 2 B2 j (mod 2 ) j=1 2 j S. Ramanujan [361] proved that~~

30B2k ≡−1 (mod 200), and that B + 4k 2 ≡ 3 (mod 5) 4k + 2 for any k ≥ 1. For new proofs of the Frobenius and Ramanujan theorems, see Howard [219]. A similar congruence is due to J. S. Frame [165]: k − 1 30B ≡ 1 + 6000 (mod 23 · 33 · 53)(94) 2k 2 Recently, E. U. Gekeler [178] has proved that for all k ≥ 4, even 2k + 24 2k 65520 ≡ − (mod 27 · 34 · 53 · 72 · 13)(95) Bk+12 Bk 691 This new result is obtained from congruences among Eisenstein series, by representing the Fourier coefﬁcients of certain combinations of Eisenstein series as Iwa- sawa functions. See also [179]. In 1998 M. Eie (see [149]) proved Kummer’s congruence (83) in a general form: Suppose that m, n are positive even integers and k is a positive integer such that (p − 1) m for all prime divisors of k. Then, if m ≡ n (mod ϕ(k)), then B B m (1 − pm−1) ≡ n (1 − pn−1)(mod k)(96) m p|k n p|k

~~(Here ϕ is Euler’s totient). For k = pN+1 one obtains (82).~~

~~544 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

The method of proof shows that one can derive congruences of the following type for values of Bernoulli polynomials. Foraﬁxed prime p > 3, let k be a positive integer not divisible by p. Suppose further that a, b are positive integers such that 1 ≤ a, b < k and a + jn = bp for some positive integer j with 1 ≤ j ≤ p − 1. Assume that (p − 1) m and that m ≡ n (mod (p − 1)pN )(N ≥ 1). Then 1 a b 1 a b B − pm−1 B ≡ B − pn−1 B (mod pN+1) m m k m k n n k m k (97) A generalization of Voronoi’s congruence (84) is due to A. Simalarides [402]: If p > 3isaprime, p a and (p − 1) 2n, then p−1 k−1 ja k−1 ( − p (p−2n)) ≡ (2n−1)p ( k), ( ) a a B(2n−1)pk +1 j mod p 98 j=1 p for each k ≥ 1. For k = 1 this gives (84), since by Kummer’s congruence

~~B(2n−1)p+1 B2n B( − ) + = [(2n − 1)p + 1] ≡ (mod p). 2n 1 p 1 (2n − 1)p + 1 2n~~

See also I. Sh. Slavutskii [410] for a proof using the methods of Voronoi. We note that applying (84) for various values of a, one can deduce congruences for Bernoulli numbers. Put e.g. a = 2, 3, 4 then adding the ﬁrst two relations, and subtracting the third one obtains:

~~B (2p−2n + 3p−2n − 4p−2n − 1) 2n ≡ j 2k−1 (mod p), p > 3 (99) 4n p/4< j~~

~~J. W. Tanner and S. S. Wagstaff [436] proved a similar, but complicated formula which holds for all p ≥ 11:~~

B (2p−2n + 9p−2n − 10p−2n − 1) 2n ≡ (1 + 22n−1 + 32n−1 + 42n−1) j 2n−1+ 4n p < < 13p 10 j 120 +(1 + 22n−1 + 32n−1 + 42n−1 + 122n−1) j 2n−1 − 32n−1 j 2k−1− 13p < < p 2p < < 7p 120 j 9 9 j 30 −(22n−1 + 62n−1) j 2n−1 − 22n−1 j 2n−1− 5p < < 17p 17p < < 3p 18 j 60 60 j 10

~~545 CHAPTER 5~~

−(22n−1 + 42n−1 + 122n−1) j 2n−1− 7p < j< 47p 18 120 −(22n−1 + 42n−1) j 2n−1 (mod p)(100) 47p < < 2p 120 j 5 n k Congruences for Tk(n) = j connected to Bernoulli numbers appear in T. j=1 Agoh [8], in connection with Fermat and Wilson quotients. For an odd prime p put Tk = Tk(p − 1). Then

~~Tk ≡ 0 (mod p) for any k ≥ 1 with(p − 1) k,(101) but Tm(p−1) ≡−1 (mod p) for any m ≥ 0 If p ≥ 3 and (p − 1) k, k even, then~~

~~2 Tk ≡ pBk (mod p ), and for p ≥ 5 with (p − 1)|k, k arbitrary, one has~~

~~2 Tk ≡ pBk (mod p )(102)~~

~~For p = 3 and k even one has ( 2) = , ≡ ( ), ≡ 3Bk mod 3 if k 2 k 0or1 mod 3 ( ) Tk 2 2 103 3Bk + 3 Bk−2 (mod 3 ) if k = 2 and k ≡ 2 (mod 3)~~

~~If p ≥ 5 and k is odd with (p − 1) (k − 1), then~~

~~2 Tk ≡ 0 (mod p )(104)~~

~~If p ≥ 3 and 1 ≤ k ≤ p − 1, then~~

~~2 Tk ≡ pBk (mod p )(105)~~

Recently A. Junod [238] has applied the p-adic mean value theorem of A. Robert [370] to prove that 0, if (p − 1) k np pB ≡ T ≡ (mod Z )(105 ) k k p − 1, if (p − 1)|k 2 p Robert’s theorem can be stated as follows: Let K be a complete ﬁeld and let (E, |· |) be an ultrametric Banach space over K . Let f (t) ∈ E[t] with coefﬁcients in E, and

~~546 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

k suppose that h, a ∈ E with |a|≤1. The Gauss norm on E[t] will be akt = E[t] {| | ≥ } | |≤| |1/(p−1) | ( + ) − ( )|≤| |· max ak : k 0 .Then, if h p , then f a h f a h f E[t]; 2 1/(p−2) h and if h is odd with |h|≤|p| , then | f (a+h)− f (a)−hf (a)|≤ f E[t], 2 with the same conclusion when p = 2, by assuming |h|≤21/2. Bn+1(t) Let now f (t) = ∈ Q [t], when f (t) = B (t), f (t) = nB − (t) and n + 1 p n n 1 apply Roberts’ theorem to deduce the above result (which, by the way implies the von Staudt-Clausen and Kummer’s classical congruences). F. T. Howard [219] has proved that if p ≥ 5isaprime and if pl 2k and (p − 1) (2k − 2), then l+3 pB2k ≡ T2k (mod p )(k > 1)

~~This is related to a result of Carlitz from 1953 [89]: If (p − 1)ps|2k (p any prime; k > 1), then~~

~~s+1 pB2k ≡ p − 1 (mod p )(105 )~~

~~By studying Fermat and Wilson quotients, as a by-product, in [8] the following is derived: For p ≥ 3 and 1 ≤ m ≤ p − 2,~~

~~p−1 p−1 B B − − 1 m + p 1 m ≡ j m (sj − )( p). ( ) − − 1 mod 106 m p 1 m p j=1 s=1~~

~~In particular,~~

− − 1 p 1 j p 1 −4B(p−1)/2 ≡ ( js − 1)(mod p), p j=1 p s=1 j where is a Legendre symbol. p We note that Tk(n) is connected to Bernoulli numbers by a kind of Euler- MacLaurin summation formula: k+1 1 k j T (n) = B + − (m + ) ( ) k − k 1 j 1 107 j=1 j j 1

~~(p − 1)! + 1 Lwt W = be the Wilson quotient. The following famous congru- p p ences are known: Let p ≥ 3beaprime. Then~~

~~547 CHAPTER 5~~

1 W ≡ B − + − 1 (mod p), (N. G. W. H. Beeger [36], M. Lerch [283]) p p 1 p Wp ≡ B2(p−1) − Bp−1 (mod p)(E. Lehmer [275]) p−2 Bk Wp ≡ (mod p)(N. Nielsen [335]) k=1 k (108) If p ≥ 5 and m ≥ 1any integer, then 1 mW ≡ B ( − ) + − 1 (mod p)(E. Lehmer [275]) p m p 1 p (109) Wp ≡ B(m+1)(p−1) − Bm(p−1) (mod p)(T. Agoh [8]) mWp ≡ B(m+1)(p−1) − Bp−1 (mod p) We note that the second relation of (109) follows from the ﬁrst one, by taking into account the well-known relation 1 B ( − ) − 2B − + 1 ≡ (mod p). 2 p 1 p 1 p H. S. Vandiver [462] proved the following two congruences: Let p ≥ 3andp a (a ≥ 1). Then

p−2 p−1 Bk p−1−k 1 Wp + qp(a) ≡ a + (a − j)(mod p) and k=1 k p j=1 (110) p−2 p−1 Bk k 1 Wp − qp(a) ≡ a + ( ja − 1)(mod p) k=1 k p j=1 The following theorem was discovered repeatedly by many authors (A. Fried- mann and J. Tamarkine [167], M. Lerch [283], N. Nielsen [335], H. S. Vandiver [462]): For p ≥ 3 and 1 ≤ m ≤ p − 2,

~~p−1 p−1 m Bm qp( j) ≡ Wp (mod p); j qp( j) ≡− (mod p). (111) j=1 j=1 m~~

a p−1 − 1 Here q (a) = (p a) denotes the Fermat quotient. p p B For the fraction m ,I.Sh. Slavutskii [409] has shown (by strengthening a the- m orem of Sylvester and Lipschitz) that for an integer a, and a positive integer m, the number ∗+ Bm a[log2 m] 1(am − 1) m

548 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS is an integer. For related results, we note that Ramanujan [361] observed that B 2m(2m − 1) m m is an integer. In fact, for arbitrary integer k, and any m ≥ 1, B km+1(1 − k2m) 2m 2m is an integer, see J. Glaisher [193]. Ramanujan proved also that for any m ≥ 0,

B + B + 2(24m+2 − 1) 4m 2 and − 2(28m+4 − 1) 8m 4 2m + 1 2m + 1 are integers ≡ 1 (mod 30). In 1991 G. Almkvist and A. Meurman [18] have shown that for any m ≥ 1, and any integers a, b, with a = 0, b am B − B m a m is an integer (here Bm(x) is the Bernoulli polynomial). B. Sury [434], K. Bartz and J. Rutkowski [34] have obtained simpler proofs. F. W. Clarke and I. Sh. Slavutskii [114] have deduced the above theorem by first proving that if p is prime and m ≥ 1, then for all integers a, b with p|a, m b a B ∈ Z( ), m a p where Z(p) denotes the ring of rational numbers whose denominator is not divisible by the prime p. C. Congruences for higher order Bernoulli numbers have been studied by many authors, including L. Carlitz [95], L. Carlitz and F. R. Olson [97], F. T. Howard [223], A. Adelberg [4], P. T. Young [488]. (z)( ) (z) = (z)( ) The Bernoulli polynomials of order z, Bm x and numbers Bn Bn 0 were first defined and studied by N. E. Norlund¨ [338] in case z = k ∈ N. These are defined by ∞ t z tn ext = B(z)(x) t − n e 1 n=0 n! Then it follows that n n (z)( ) = (z) n− j Bn x B j x j=0 j

~~549 CHAPTER 5~~

( , ) = (n+s+1)( + ) Let An x s Bn x 1 be the so-called Narumi polynomials. Let p be a prime, and let l, k, a, b be p-adic integers such that l ≡ k (mod pα) and a ≡ b (mod pα).Ifn < p − 1, then (see [4])

~~α An(a, l) ≡ An(b, k)(mod p )(112)~~

< − (l) In particular, if n p 1, and l is a p-adic integer, then Bn is a p-adic integer, ≡ ( α) (l) ≡ (k) ( α) < < − and if l k mod p , then Bn Bn mod p , while if 0 n p 1 then | (l) (l)/ ≡ (k)/ ( α) < < − (l)/ ≡− / l Bn and Bn l Bn k mod p .If1 n p 1, then Bn l Bn n (mod l), with sign ”+”ifn = 1. (113) For n = p − 1 one has

~~pBp−1(l) ≡−l (mod pl)(114)~~

In what follows, suppose that l is a p-adic integer, and put r = [n/(p − 1)]∗, σ = − ( − ) (z) = n r p 1 . Assume that B−1 0. Then the following congruences are true (see [4]): pr B(l) n + r − l B(l) (−1)n n ≡ (−1)σ+r σ (mod p). (115) n! r σ! If p > 2andr ≤ 1 then (− )n pr B(l) 1 n ≡ n ! + − − − + − σ σ +r−1 n r l 1 r 1 n r l (l−rp) p (l−rp) ≡ (−1) Bσ − B + r − 1 σ! r 2 σ−1 p ( − + ) +(−1)p−1 B l rp p (mod p2)(116) (p + σ − 1)! p+σ −1 If p = 2andn = 1, then 2n − l − 1 2n2 + l (−1)m2n B(l)/n! ≡ (−1)n−1 (mod 4)(117) n n − 1 n

If p (n − l) then we can derive stronger congruences mod pl and mod p2l in (z)( ) terms of low degree polynomials Bm 1 . ( , ) = (− )n r (n−k)/ Let Ak p n 1 p Bn n! Then for l = n − k, relation (115) can be rewritten as r + k B(σ−k−r) A (p, n) ≡ (−1)σ +r σ (mod p)(118) k k σ! Forsmall values of σ , (118) contains results due to Howard [223]. We quote also the following theorem by Adelberg:

~~550 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~σ> | r (l)/ If p n or 0, then l p Bn n!; (119) If σ>0andp n or σ>1, then B(l) n + r − l B(l) (−1)n pr n ≡ (−1)σ+r σ (mod pl)(120) n! r σ!~~

~~(If σ = 0 and p n, multiply right hand side of congruence by l/(l − r − n)); If p|l and σ>1, then (l) B n + r − l l Bσ (−1)n pr n ≡ (−1)σ +r−1 · (mod pl)(121) n! r σ σ!~~

~~(If p|l, σ = 1 and p n, change sign of right hand side). Foracorollary, suppose that σ = sp(n) = sum of digits of n in base p (see Chapter 4) and σ~~

| (l),() l Bn 122 If σ>1, then n + r − l n! B(l) B(l) ≡ (−1)r · σ (mod pl), (123) n r pr σ! If p|l and σ>1, then n + r − l n! l Bσ B(l) ≡ (−1)r−1 · · (mod pl)(124) n r pr σ σ! We note that (119) and (122) strengthen a theorem of Howard [223] (by weaken- ing the hypothesis in his theorem from p k to p k or (p − 1) n), while Howard’s theorem improves an earlier result by Carlitz. Another corollary says that if 1 <σ

~~( ) l B l ≡ B(l) ≡− (mod pl)(125) p+σ −1 σ p σ In particular,~~

~~(σ p) (σ p) 2 B ≡ Bσ ≡−pBσ (mod p ), and p+σ−1 p ( ) (p) (p) p 2 126 Bσ ≡ B ≡− Bσ (mod p ) p p+σ−1 σ The following result: If p is an odd prime and p|l, then~~

~~(l) ≡ 2/ ( 2)() Bp l 2 mod pl 127~~

~~551 CHAPTER 5 generalizes a Carlitz congruence [95], namely:~~

~~(p) ≡ 2/ ( 3)() Bp p 2 mod p 128~~

~~r ( ) , Forthe values of higher order Bernoulli polynomials Bn x (n r positive integers) L. Carlitz [83] proved that for r ≥ 1, p odd prime and a ∈ Zp~~

t(r) (r)( ) − , p Bn a is a p adic integer t(r) denoting the number of nonzero digits of r in the p-adic expansion (and n ≥ 0). A. Junod [238], by applying the mean value theorem of Robert [370] has recently proved that for all m, n ≥ 0, a ∈ Zp,

r (r) r (r) np p B + (a) ≡ p B + (a)(mod Z ), m np m n 2 p and np pr B(r)(t) ≡ pr B(r)(t p)(mod Z [t]) np n 2 p The universal Bernoulli numbers Bn have been introduced in 1989 by F. W. Clarke [112] as follows: Let G(t) be the compositional inverse of F(t) = ∞ i+1 ci t /(i + 1) (a power series over the polynomial ring Q[c1, c2,...], ci being i=0 indeterminates). Then ∞ t = B tn/n ( ) ( ) n ! 129 G t n=0

i t Note that for ci = (−1) , F(t) = log(1 + t), G(t) = e − 1, B)n ≡ Bn, the ordinary Bernoulli numbers. The following Kummer type congruences are due to A. Adelberg [5]: If n = q(p − 1) + 1 > 1 (i.e. n ≡ 1 (mod (p − 1))), then Bn 1 q q p q−1 q−1 1 − q q−2 ≡ c c + c c − c c + c c − (mod p) ≡ n 2 1 p−1 2 1 p−1 p−1 p 2 p−1 2p 1

1 q 1 − n p q−1 q−1 n q−2 ≡ c c + c c − c c + c c − (mod p)(130) 2 2 p−1 2 1 p−1 p−1 p 2 p−1 2p 1 Similar congruences when n ≡ 0, 1 (mod (p − 1)) are proved in [6]. In [5] there are deﬁned also universal poly-Bernoulli numbers Bn,k of index k by means of the polylogarithm function, and universal von Staudt-Clausen type congruences are deduced.

~~552 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

In 1991 I. Dibag [137] considered for a given set S of primes the series L S(x) = (−1)n−1xn/n, where the prime factors of n belong to S. Let the rational numbers n (dn) be deﬁned by ∞ xn / −1( ) = ( ) x L S x dn 131 n=0 n! −1( ) ( ) where L S x denotes the functional inverse of L S x . Dibag’s von Staudt-Clausen theorem states that if p ∈ S and (p − 1)|2n, then d2n =− 1/p (mod Z), which p may be formulated also as

~~pd2n ≡−1 (mod pZ(p)) if p ∈ S,(p − 1)|2n (132) where Z(p) denotes the ring of p-local integers. F. W. Clarke [113] proves that if m is even and (p − 1)|m, and p ∈ S, then~~

1+γp(m) pdm ≡ p − 1 (mod p Z(p)), otherwise γp(m) dm ≡ 0 (mod p Z(p)), (133) γ ( ) where p p n n.For a similar analogue of the von Staudt-Clausen theorem, see I. Dibag [138]. D. Forapplication in the theory of general abelian number fields, generalized Bernoulli numbers belonging to residue class characters were first introduced by N. C. Ankeny, E. Artin and S. Chowla [27] in the special case of a quadratic character, and by H. W. Leopoldt [280], [281] in general. Let χ be a primitive character with n conductor f . Then the generalized Bernoulli numbers Bχ are defined by

~~∞ f χ(a)teat = Bn tn/n! (134) ft − χ a=1 e 1 n=0~~

n If f = 1, i.e. if χ is the principal character (χ(a) = 1 for all a), then Bχ ≡ Bn for 1 1 n all n = 1, and Bχ = =−B . The numbers Bχ are algebraic numbers and belong 2 1 to the smallest algebraic number ﬁeld containing all χ(a).The generalized Bernoulli polynomials are deﬁned by n n n j n− j Bχ (x) = Bχ x (135) j=1 j

~~553 CHAPTER 5~~

~~n n Then we have Bχ (0) = Bχ , and it can be shown that they are connected with the classical Bernoulli polynomials via f + n n−1 a x Bχ (x) = f χ(a)Bn (136) a=1 f~~

n Kummer type congruences for Bχ were ﬁrst proved by L. Carlitz in 1959 (see n [92]). For von Staudt-Clausen and Voronoi type congruences for Bχ see T. Agoh [10], I. Sh. Slavutskii [407], L. C. Washington [479]. In 1993 K. Girstmair [191] has n introduced the Bernoulli numbers Bg belonging to g, where g is a periodic mapping from Z to C with period m (m positive integer, n nonnegative integer), as follows:

~~m ∞ ( ) kt /( mt − ) = n n/ ( ) g k e t e 1 Bg t n! 137 k=1 n=0~~

= χ − n ≡ n ( ) = 2kiπ/m When g a primitive character, then Bg Bχ .When g k e , the cyclotomic Bernoulli numbers of Girstmair are reobtained (see [190]). Girstmair n proves von Staudt-Clausen and Kummer type congruences for Bg .For a Kummer congruence for g = χ = ωk, where ω is the Teichmuller¨ character mod p (p odd prime), see H. S. Gunaratne [203]. If χ is the (unique) quadratic character with conductor f = 4, i.e. χ(1) = 1, χ(3) =−1, χ(2) = χ(4) = 0, then it follows immediately that

~~2 n+1 E =− Bχ (138) n n + 1 1 for the Euler numbers E . Since E = 2n E , this follows from a more general n n n 2 property of Euler polynomials (see B. C. Berndt [45])~~

1 1−n n+1 E (x) =− 2 Bχ (2n − 1)(139) n n + 1 Therefore many properties of Euler numbers and polynomials follow also from n n the corresponding properties of the generalized numbers Bχ and polynomials Bχ (x). n Forageneralization of the Polya-Vinogradov´ inequality, involving Bχ ,see S. Kane- n mitsu and K. Shiratani [242]. For congruences for Bχ belonging to unequal characters,see I. Sh. Slavutskii [406]. E. The sequence of (classical) Bernoulli numbers is periodic after being reduced a modulo n (n ≥ 1). The rational number with b > 0, (a, b) = 1issaid to be n- b integral if (b, n) = 1 (i.e. iff the congruence yb ≡ a (mod n) has a unique solution

~~554 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

y ∈{0, 1,...,n − 1}). By the von Staudt-Clausen theorem B2i is p-integral (where p isaprime)iff(p − 1) 2i. Let L(n) be the smallest integer > 1 with the following property: there exist m0 such that for all k, and m ≥ m0

~~Bk n − integral and k ≡ m (mod L(n)) ⇒~~

~~Bm n − integral and Bk ≡ Bm (mod n)(140)~~

L(n) is called the period-length of the sequence (Bk (mod n)),while the smallest m0 in (140) will be called the preperiod of (Bk (mod n)), and denoted by V (n). The properties of L(n) and V (n) have been studied in detail by W. Herget [216]. Since (n1, n2) = 1 ⇒ L(n1, n2) = lcm[L(n1), L(n2)] and B(n1n2) = α max{V (n1), V (n2)},itissufﬁcient to study the case n = p (p prime, α ≥ 1). One has: L(2α) = L(3α) = 2; V (2α) = V (3α) = 2, (141) L(pα) = pα(p − 1) for p ≥ 5; V (pα) ≤ α + 1 The last relation of (141) can be improved as follows: V (p) = 2 for p = prime.

Let p ≥ 5beaprime and α ≥ 2even. Then if Bα ≡ 0 (mod p) and (p − 1) α, the V (pα) = α + 1. Let k be the greatest integer such that for all 0 ≤ i ≤ k one has Bα−2i ≡ 0 (mod p2i+1) and (p − 1)|(α − 2i). Then

~~V (pα) = α − 1 − 2k (142)~~

2 Similarly, if p ≥ 5 and α ≥ 3isodd, then if Bα−1 ≡ 0 (mod p ), and (p − 1) (α − 1), then V (pα) = α. Let k be the greatest integer such that for all 0 ≤ i ≤ k one has Bα−1−2i ≡ 0 (mod p2i+2) and (p − 1)|(α − 1 − 2i).Then V (pα) = α − 2 − 2k (143) The p-divisibility of certain sets of Bernoulli numbers has been studied by S. V. Ullom [459]. For a prime p ≥ 5, let I ={n ∈ N : n even and 2 ≤ n ≤ p − 3}. p − 1 For all even divisors d of p − 1 for which is odd, set I (d) ={2k ∈ I : d (2k − 1, p − 1) = (p − 1)/d}. Then ϕ(d) log log p card{m ∈ I (d) : p|B } < + ϕ(d) ,(144) m 2 log p ϕ being Euler’s totient. See also S. S. Wagstaff, Jr. [472].

~~555 CHAPTER 5~~

The distribution of Bernoulli numbers modulo primes has been considered by W. L. Fouche[´ 164]. Let p ≥ 3beaprime. Let A be any set of n consecutive integers in (0, p), and suppose that 2 ≤ t ≤ p − 1. Then

~~9 1/3 2/3 card{n ∈ A : B − (n) ≡ α(mod p)}≤ t m (145) p t 2 for every integer α. Furthermore, " p−2 p−1−k 9 2/3 card n ∈ A : Bkn ≡ α(mod p) ≤ m (146) k=1 2~~

~~It is well known that for every prime p, pBp−1 ≡−1 (mod p).T.Agoh [9] conjectures that~~

nBn−1 ≡−1 (mod n) ⇔ n prime (147) This has been veriﬁed up to n ≤ 49999 (see [481]). B. C. Kellner [246] provided a short proof of Agoh’s result on the equivalence of conjecture (147) to a famous conjecture of Giuga (see [192], [58]): If n > 1, then

~~n−1 kn−1 ≡−1 (mod n) ⇔ n prime (148) k=1~~

~~The Giuga-Agoh conjecture states that n/p ≡ 1 (mod n) ⇔ n prime (149) p|n,(p−1)|(n−1)~~

The factorization of Bernoulli numerators goes back to M. Ohm [340]. Then J. C. Adams [3] in 1878 published a table for the ﬁrst sixty-two numbers of Bernoulli. In 1978 S. S. Wagstaff [473] published the factorizations through N60 and E42.Recently [474] he completely factored all N2k for 2k ≤ 152, and also the Euler numbers E2k for 2k ≤ 112. The results are completed continuously at the web address [475]. F. The prime factors of the Euler numbers determine the structure of certain cyclotomic ﬁelds; see e.g. R. Ernwall and T. Metsankyl¨ a¨ [157]. Since the Euler numbers are all integers, there is no analogue for them of the von Staudt-Clausen theorem. But Kummer’s congruence has an analogue, due also to Kummer [265]: If n ≥ 1 and p ≥ 3isaprime, then

~~E2n+p−1 ≡ E2n (mod p)(150) See also L. Carlitz and J. Levine [96] and S. S. Wagstaff, Jr. [474]. The following two theorems have been discovered recently by Wagstaff [474], the ﬁrst being an~~

~~556 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS analogue of Adams’ theorem (83): Let p be an odd prime, n ≥ 1 and N ≥ 0an integer. Suppose (p − 1)pN |n. Then~~

~~N+1 En ≡ 0or (mod p )(151) according as p ≡ 1or3 (mod 4). For all integers n ≥ 0 and k ≥ 0wehave~~

~~≡ + k ( k+1)() E2n E2n+2k 2 mod 2 152~~

~~Theorems of type (151) and (152) were probably known to Kummer, too. In 1938 E. Lehmer [275] proved that if p ≡ 5 (mod 8) is a prime, then~~

~~E(p−1)/2 ≡ 0 (mod p)(153)~~

Foranelementary proof, see R. Ernvall [155]. In 1982 R. Ernvall [156] proved that (153) holds true for all p ≡ 1 (mod 4). For connections of congruences of type (153) with Diophantine equations (Pell equations), see L. J. Mordell [314]. In [84] Carlitz proves that none of the following numbers

~~|E4n|, |E8n+6|, |E24n+18|, |E40n+34|, |E40n+26| (154) is a perfect square, and conjectures that the same is true for~~

~~|E8n+2| (155)~~

(i.e. |E2m| is never a square for m > 1). In the proof of (154) various congruences are used, e.g. E2n ≡ 1 − 2n (mod 8) for all n ≥ 1 (156) The congruence E4n ≡−1 (mod 3)(157) appears in N. Nielsen [334]. On the other hand

~~E4n ≡ 1 (mod 4), (158)~~

∗ ∗ = (− )n ∗ = More generally, let us deﬁne the numbers En by E2n 1 E2n, E2n+1 0 ( ≥ ) ( ) ∗ = ∗ ( ≥ ) n 0 (see 12 ). Put Em E2m m 0 . Then A. Junod [238] has proved that ∗ ≡ ∗ ( ), ∗ ≡ ∗ ( ), Em+2 Em mod 4 Em+3 5Em+1 mod 36 ∗ ≡ ∗ − ∗ ( ), ( ) Em+4 14Em+2 9Em mod 576 159 ∗ ≡ ∗ − ∗ ( ) Em+5 30Em+3 89Em+1 mod 14400

557 CHAPTER 5 and that ∗ ≡ ∗ ≡ ···≡ ∗ = ( ), Em+1 Em E0 1 mod 4 E∗ ≡ 5E∗ ≡ ···≡5m E∗ = 5m+1 (mod 36), m+2 m+1 2 (160) 1, if m is even E∗ ≡ E∗ ≡ ···≡ (mod 30) m+3 m+1 5, if m is odd

~~These follow from more general results for the Meixner polynomials Qn(x) de- 2 ﬁned by Q0(x) = 1, Q1(x) = x, Qn+1(x) = xQn(x) − n Qn−1(x)(n ≥ 1):Ifp is an odd prime, then~~

~~p (p−1)/2 Q p(x) ≡ x − (−1) x (mod pZp[x])(161)~~

~~Junod conjectures that~~

~~( ) ≡ n ( ) ≡ ( p − (− )(p−1)/2 )n ( Z ), ( ) Qnp x Q p x x 1 x mod np p[x] 162 which would imply~~

~~∗ ≡ (− )n(p−1)/2 ∗ ( Z ) Em+np 1 Em+n mod np p and ( ) ∗ ≡ ∗ ( Z ) 163 Em+np Em+n mod np p for any m, n ≥ 0, and p = 2aprime. The higher order Euler polynomials are deﬁned by~~

~~∞ 2 r tn ext = E(r)(x) (164) t + n e 1 n=0 n!~~

~~, ≥ = (1)( ) = ( ) where n r 0 are integers. For r 1, En x En x ,asgiven by relation (14). By generalizing results by M. Zuber [489], A. Junod [238] has recently proved that~~

(r)( ) ≡ (r)( p)( Z )() Enp t En t mod np p[t] 165 for all integers n, r ≥ 0, and an odd prime p.Forr = 1 this gives a Honda type congruence for the classical Euler polynomials. Further, if a ∈ Zp, then (r) ( ) ≡ (r) ( )( Z )() Em+np a Em+n a mod np p 166 , , ≥ ( (r)( )) − for all n m r 0. Particularly, the sequence En a n≥1 is p 1-periodical (mod pZp): (r) ( ) ≡ (r)( )( Z )() Em+p−1 a Em a mod p p 167 As for the variant of Euler numbers deﬁned in (159), there are also other ”Euler numbers” deﬁnition in the literature. However, we wish to use always the classical

~~558 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS deﬁnition (11) for the Euler numbers. For example, in [380] the Euler numbers are deﬁned by ∞ xn En = tan x + secx (168) n=0 n!~~

n Then clearly, E2n = (−1) E2n (called also as ”secant numbers”), and E2n+1 = B + (−1)n 2n 2 · 4n(4n+1 − 1) (called also as ”tangent numbers”). In [420] R. Stanley 2n + 2 has shown that the volume of the convex polytope determined by xi ≥ 0 (i = 1, n), and xi + xi+1 ≤ 1 (1 ≤ i ≤ n − 1) is given by En.(169) In fact, En counts the number of permutations π = a1a2 ...an in Sn (the symmetric group of all permutations of {1, 2,...,n}) that alternate, i.e. a1 < a2 > a3 < ..., due to D. Andre´ [24] (see also [116]). There are many known q-analogs of Euler numbers or its variants. See e.g. a paper by L. Carlitz [82] from 1954. The divisibility properties of q-tangent numbers have been studied e.g. by G. Andrews and I. Gessel [26], D. Foata [162] etc. Let q ≥ 1, and as usual put [k] = k−1 [k]q = 1 + q + ···+q . The q-Euler numbers En|k(q) can be deﬁned (see Sagan and Zhang [380]) by the recurrence

~~∗ [n/k] n n−mk+1 E( + )| (q) = q E( − )| (q)E( − + )| (q)+ n 1 k − mk 1 k n mk 1 k m=1 mk 1~~

+χ(k n)En|k(q), (n ≥ 0)(169) ( ) = χ( ) and E0|k q 1, where P is the characteristic function of property P (i.e. is 1 if n [n]! P is true, and 0 if it is false), and = ([n]! = [n][n − 1] ...[1]) is m [k]![n − k]! the Gauss q-binomial coefﬁcient. Sagan and Zhang have proved the following divisibility theorems: Let p be a prime, and 1 ≤ i ≤ p − 1. Then

~~n E(np+i)|p(q) is divisible by [p] (170)~~

~~Further, ( ) ... ( ) E(np+i)|p q is divisible by [p][p]q2 [p]q3 [p]qn 171 For p = 2, i = 1 these were proved in [26]. Let En|k = En|k(1). Then (170) gives:~~

~~n n E(np+i)|p is divisible by [p] = p (172)~~

~~559 CHAPTER 5 for all 1 ≤ i ≤ p − 1. I. Gessel and G. Viennot [189] have shown that~~

∗ np− j − np p p 1 E( − )| (173) j np j p where [·]∗ denotes the integer part. For p = 2, j = 1 this reduces to the well known fact that 2n−1 2 |2nE2n−1 (174) where E2n−1 = E2n−1|2 is a (classical) tangent number. In 1911 F. H. Jackson [235] deﬁned the q-variant of sin, cos and tan by ∞ ∞ (−1)n x2n+1 (−1)n x2n (x) = , (x) = sinq + cosq n=0 [2n 1]! n=0 [2n]! (175) 1 tanq (x) = sinq (x)/ cosq (x), secq (x) = cosq (x)

~~Then the q-Euler numbers En(q) may be deﬁned by xn E (q) = (x) + (x)(n ≥ ) n ( , ) tanq secq 0 n≥0 q q n~~

~~n−1 l where (x, q)n = (1 − xq ), for n ≥ 1and(x, q)0 = 1. l=0 More generally, J. Desarm´ enien´ [133] deﬁned~~

sinq (u; x) = sin u cosq (x) + cos u sinq (x) = um xn = (− )(m+n−1)/2 (m, n ≥ , m + n = ) 1 ( , ) 0 odd m,n m! q q n and cosq (u; x) = cos u cosq (x) − sin u sinq (x) = um xn = (− )(m+n)/2 (m, n ≥ , m + n = ) 1 ( , ) 0 even m,n m! q q n 1 tanq (u; x) = sinq (u; x)/ cosq (u; x), secq (u; x) = cosq (u; x)

~~He deﬁned the numbers Em,n(q) by m n u x E , (q) = (u; x) + (u; x). ( ) m n ( , ) tanq secq 175 m,n≥0 m! q q n~~

~~560 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~One has E0,n(q) = En(q), Em,n(1) = Em+n. The following congruences are true: Let m, k, a, b be integers, m, a, b ≥ 0, k ≥ 1, Then: i) if k is even, and m + ka + b is odd, with a ≥ 1, then~~

~~Em,ka+b(q) ≡ 0 (mod k(q)).~~

~~If m + ka + b is even, then~~

~~(k+2)a/2 Em,ka+b(q) ≡ (−1) Em,b(q)(mod k(q)) (175 )~~

~~ii) If k is odd, then~~

(k−1)a/2 Em,ka+b(q) ≡ (−1) Em+a,b(q)(mod k(q)), where k denotes the kth cyclotomic polynomial (see Chapter 3). For m = 0, relation i) of (170 ) is due to G. Andrews and I. Gessel [26], see also [162] and [25]. For k = 2, q = 1 these were proved by Kummer in 1850. In 2000 H. Prodinger [356] as variations to Jackson’s deﬁnitions, has deﬁned the q-trigonometric function

∞ n 2n+1 ∞ n 2n (−1) x 2 (−1) x 2 sin∗(x) = qn , cos∗(x) = qn , q [2n + 1]! q [2n]! n=0 n=0 (176) ∗( ) = ∗( )/ ∗( ) tanq x sinq x cosq x and deduced among others divisibility properties of these tangent numbers. For continued fraction expansions, see M. Fulmek [169]. For another q-analogue of the Euler numbers, see G.-N. Han, A. Randrianarivony and J. Zeng [212]. G. Several nonequivalent deﬁnitions of q-analogues of Bernoulli polynomials and numbers are known in the literature. In 1948 L. Carlitz [91] deﬁned the q- Bernoulli numbers βk(q) by k k + 1, k = 1 β (q) = 1, q j 1β (q) − β (q) = (177) 0 j k 0, k ≥ 2 j=0 j

~~In 1954 he introduced the q-Euler numbers Hk(u, q) (associated to u)by (see [82]) k k j H0(u, q) = 1, q H j (u, q) − uHk(u, q) = 0 (k ≥ 1)(178) j=0 j~~

~~561 CHAPTER 5~~

Note that if q → 1, then βk(q) → Bk and Hk(u, q) → Hk(u), where (Bk) are the ordinary Bernoulli numbers, while Hk(u) are defined by ∞ 1 − u tk = H (u) (179) t − k e u k=0 k! J. Satoh [392] defined a q-Riemann ζ-function by ∞ ∞ 2 − s qn qn ζ (s) = (1 − q) + (s ∈ C)(180) q − s−1 s s 1 n=1 [n] n=1 [n] and proved that ζq is analytic in the whole complex plane. For k ≥ 1, integer, one has   qβ1(q) if k = 1 ζ (1 − k) = β (q) (181) q  − k if k ≥ 2 k Further, Satoh defined ∞ u−n l(s, u, q) = (s ∈ C) s n=1 [n] and proved that l(s, u, q) is analytic in C, and for k ≥ 0 integer   1  if k = 0 (− , , ) = u − 1 ( ) l k u q  u 182  H (u, q) if k ≥ 1 u − 1 k Foraconstruction of q-Bernoulli type numbers using Stirling numbers, see J. Satoh [390]. The q-Bernoulli polynomials βk(x, q) can be defined inductively by

~~x k βk(x, q) = (q β + [x])~~

i with replacing β by βi (x, q). N. Koblitz [257] deﬁned the generalized q-Bernoulli numbers βk,χ (q), where χ is a primitive Dirichlet character with conductor f : f k−1 a a f βk,χ (q) = [ f ] χ(a)q βk , q (183) a=1 f

~~Sato [392] has constructed the generalized q-Euler numbers Hk,χ (u, q) by f k f −a f a f Hk,χ (u, q) = [ f ] u χ(a)Hk u , , q (184) a=1 f~~

~~562 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~where the q-Euler polynomials Hk(u, x, q) are deﬁned inductively by~~

~~x k Hk(u, x, q) = (q H + [x]) (k ≥ 1)~~

i with replacing H by Hi (u, x, q). Further, Sato introduced a q − L-series Lq (x,χ) by ∞ ∞ 2 − s qnχ(n) qnχ(n) L (s,χ)= (q − 1) + ,(185) q − s−1 s s 1 n=1 [n] n=1 [n] and a function lq (u, s,χ)by

~~∞ u−nχ(n) l (u, s,χ)= ( ) q s 186 n=1 [n] These functions interpolate the q-Bernoulli and q-Euler numbers as follows:~~

Lq (1 − k,χ)=−βk,χ (q)/k, 1 (187) l (u−, −k,χ)= H ,χ (u, q), q 1 − u f k for k > 1 integer. The values Lq (k,χ)for k ≥ 2 integer have been evaluated by H. Tsumura [453]: (−1)k 1 f [a] L (k,χ)= · qa(2−k)χ(a)G(k) q−a , q f ,(188) q ( − ) k k 1 ! [ f ] a=1 [ f ] where ∞ 1 (−1)k+1 1 G(x, q) = x − log x − x + β + (q) + ( + ) k 1 q 1 q k=1 k k 1 x for |x| > 1/(1 −|q|)2 (with |q| < 1). Since ∞ ∞ (−1)k 2 − k qn qn Gk(x, q) = (1 − q) + (1 + x − qx) ( − ) − ( + )k−1 ( + )k k 1 ! k 1 n=0 x [n] n=0 x [n] for k ≥ 2 integer, one has a q-analogue of the classical formula for the Euler gamma function: ∞ (−1)k dk 1 log (x) = ( − ) k ( + )k k 1 ! dx n=0 x n Therefore, G(x, q) can be considered as a q-log- function.

~~563 CHAPTER 5~~

~~In 1991 H. Tsumura [389] deﬁned the modiﬁed q-Bernoulli numbers βn(q) by~~

∞ n t Fq (t) = βn(q) ,(189) n=0 n! where Fq (t) is determined as a solution of the following q-difference equation: q − 1 F (t) = et F (qt) − t, F (0) = q q q log q t Moreover, we let F (t) = , and β (1) = B , which is the ordinary 1 et − 1 n n Bernoulli number. For 0 < q < 1 the following series representation is valid: − ∞ q 1 t/(1−q) n [n]t Fq (t) = e − t q e log q n=0

~~Thus, Fq (t) is continuous as a function of (q, t) on (0, 1] ×{t ∈ C : |t| < 2π}. The modiﬁed q-Bernoulli polynomials βn(x, q) are deﬁned by~~

∞ n x [x]t t Fq (q t)e = βn(x, q) (190) n=0 n! Note that − ∞ x [x]t q 1 t/(1−q) n+x [n+x]t Fq (q t)e = e − t q e log q n=0 J. Satoh [389] considered the q-series ∞ n s Zq (s) = q /[n] n=1 which interpolated (βk(q)) at non-positive integers for a complex number q with |q| < 1, and where [x] = (1 − q x )/(1 − q).Let

s Zq (s) = Zq (s) + (1 − q) /((1 − s) log q). Note that if q → 1, then Zq (s) → ζ(s) for Re s > 1. Satoh proved that   β ( ) − k q ≥ Z (1 − k) = if k 2 (191) q  k −β1(q) − 1ifk = 1

~~564 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

In [452] H. Tsumura considered Zq (s) at positive integers s.Particularly, we quote the following result: ∞ ∞ qn cos([n]π) qn sin([n]π) 2kZ (2k + 1) − 2k − π = q 2k+1 2k n=1 [n] n=1 [n]

− k 1 (−1) j−1π 2 j (1 − q)2k+1 π = ( k − j)Z ( k − j + ) − + ( ) 2 2 q 2 2 1 cos − j=1 2 j ! log q 1 q ∞ (−1)mπ 2m +(− )k+1π 2k β (q), ( ) 1 ( + ) 2m 192 m=0 2k 2m ! which for q → 1gives a result of D. Cvijovic´ and J. Klinowski [126]: − (−1)k(2π)2k k 1 (−1) j−1 j ζ(2 j + 1) ζ(2k + 1) = · + ( k+1 − ) ( − ) π 2 j k 2 1 j=1 2k 2 j ! ∞ (2m)! ζ(2m) + · (193) ( + ) 2m m=0 2m 2k ! 2 The modified q-Hurwitz ζ-function is defined by ∞ (1 − q)s ζ (s, x) = + qn+x / n + x s x > . q ( − ) [ ] for 0 1 s log q n=0 Then for k ∈ N, one has β (x, q) ζ (1 − k, x) =− k q k The modified q − L-series and generalized q-Bernoulli numbers are given by f −s a Lq (s,χ)= χ(a)[ f ] ζq f s, a=1 f and f k−1 a f βk,χ (q) = [ f ] χ(a)βk , q (k ≥ 0) a=1 f n Note the lim βk,χ (q) = Bχ , the generalized Bernoulli numbers given by (134). q→1 The following formula holds true: β ,χ (q) L (1 − k,χ)=− k (k ≥ 1 integer)(194) q k

~~565 CHAPTER 5~~

For (194) and related results, see H. Tsumura [455]. By applications of methods of [455] the following result of M. Katsurada [245] (who used Mellin transform) can be reobtained: Let n be a positive integer, x a real number with |x|≤1, and let f ∞ τ(χ) = χ(a) exp(2πia/f ) be the Gauss sum. Let L(m,χ) = χ(γ)γ−m be a=1 γ =1 the Dirichlet L-function. If χ(−1) = 1 and χ ≡ 1, then ∞ ∞ χ(l) cos(2πlx/f ) πx χ(l) sin(2πlx/f ) nL(2n + 1,χ)− n − = 2n+1 2n l=1 l f l=1 l " − ∞ 2πx 2n n 1 (−1)k−1kL(2k + 1,χ) τ(χ) (2k)!L(2k, χ) = (−1)n + x2k ( − ) ( π / )2k ( + ) f k=1 2n 2k ! 2 x f f k=1 2n 2k ! If χ(−1) =−1, then ∞ χ(l) cos(2πlx/f ) L(2n,χ)− = 2n l=1 l − − 2πx 2n 1 n 1 (−1)k−1 L(2k,χ) = (−1)n + ( − ) ( π / )2k−1 f k= 2n 2k ! 2 x f 1 " ∞ 2iτ(χ) (2k)!L(2k + 1, χ) + x2k+1 ( ) ( + ) 195 f k=0 2n 2k ! For q-analogue of the p-adic L function of Kubota and Leopoldt [264], and connections with q-Bernoulli and Euler numbers, see N. Koblitz [257], H. Tsumura [456]. For the q-extension of Morita’s p-adic gamma function, see N. Koblitz [258], Y. S.Kim [250]. For p-adic log-gamma functions, see e.g. J. Diamond [136], T. Kim [249]. See also Y. Morita [315]. For the q-gamma functions of F. H. Jackson, see R. Askey [29], D. S. Moak [313], M. Badiale [31], A. Gupta [204], A. B. Olde Daal- huis [341], K. Ueno and M. Nishizawa [458], H. Alzer [21], J. W. Son and D. S. Jang [415]. For p-adic functions attached to the Lubin-Tate groups, see K. Shiratani [400], K. Kozuka [127] and H. Tsumura [454]. By means of p-adic integrals with respect to the q-analogue of the p-adic measure defined by T. Kim [249], recently J. W. Son and M. S. Kim [252] have defined ∗( ) ∗( , ) other p-adic q-analogs Bn q and Bn x q of Bernoulli numbers and polynomials. ∗( ) The numbers Bn q can be defined by ∞ q − 1 log q + t tn = B∗(q) ,(196) t − n log q qe 1 n=0 n!

~~566 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

∗( ) while the q-analogs Hn q of Euler numbers are defined by ∞ q − 1 tn = H ∗(q) (197) t − n qe 1 n=0 n! ∗ ( ) ∗ ( ) χ They defined also Bn,χ q and Hn,χ q , where is a Dirichlet character, proved ∗ ( ) ∗( ) Kummer type congruences for Hn,χ q , and some relations between Bn q and ∗( ) Hn q . H. The Euler numbers (En) should nor be confused with another kind of numbers A(n, k), introduced by Euler in 1755, and called Eulerian numbers. These numbers may be defined recursively via

~~A(n + 1, k) = (n − k + 2)A(n, k − 1) + kA(n, k), (k ≥ 2, n ≥ 0)(198)~~

~~A(n, 1) = 1 for n ≥ 0 and A(0, k) = 0 for k ≥ 2. These numbers are generated by~~

1 − u n tn G = G(t, u) = = 1 + A(n, k) uk−1 t(1−u) − e u k=1 n! It is easy to verify that ∂G ∂G (u − u2) + (tu − 1) + G = 0, ∂u ∂t and by letting the coefﬁcient of uk−1tn/n!equal to 0 in this differential equation, (198) follows at once. Some authors deﬁne the Eulerian numbers a(n, k) which satisfy

a(n, k) = (n − k)a(n − 1, k − 1) + (k + 1)a(n − 1, k)(n ≥ 1, k ≥ 1)(199) with a(n, 0) = 1 for n ≥ 0 and a(0, k) = 0 for k ≥ 1. n The notation a(n, k) = is also applied. (We note that some authors use also k n A(n, k) = ). These numbers a(n, k) count in fact the number of permutations π k of the symmetric group Sn of {1, 2,...,n} having exactly k ascents (an ascent is an occurrence of π(j)<π(j + 1) for 1 ≤ j ≤ n − 1), see e.g. [116]. However, the numbers A(n, k) and a(n, k) are connected by

~~A(n, k) = a(n, k − 1)(k ≥ 2)(200) and they have the symmetric property~~

~~A(n, k) = A(n − k + 1), a(n, k) = a(n, n − k − 1)(201)~~

~~567 CHAPTER 5~~

Therefore, any property for A(n, k) can be written via (200) to a property of a(n, k) and vice-versa. Twofundamental properties of A(n, k) are given by: k n + 1 A(n, k) = (−1) j (k − j)n,(202) j=0 j and the Worpitzky identity from 1883 (see [116] and [487]) n x + k − 1 xn = A(n, k) (203) k=1 n

~~The Eulerian polynomials An(x) are deﬁned by~~

~~n−1 k+1 An(x) = A(n, k + 1)x , n ≥ 1 (A0(x) = 1) k=0~~

~~While no closed formula is known for the Eulerian polynomials, their generating function is given by (see e.g. D. Foata and M. P. Schutzenberger¨ [163])~~

∞ A (x)tn 1 − x n = and − (1−x)t n=0 n! 1 xe (204) ∞ ( ) n − An x · t = 1 ( − )n+1 − −t n=0 x 1 n! 1 xe giving the recurrence relation n−1 n−1 n n−1−i An(x) = x(1 − x) + Ai (x)x(1 − x) i=1 i

The Eulerian numbers A(n, k) and a(n, k) occur in many ﬁelds. For example, F. Schmidt and R. Simion [395] deal with a set of probability problems that lead to the computation of volumes of regions obtained by dissecting a polytope P with hyperplanes. They consider in particular the case when P is an n-dimensional symplex. The regions formed by dissecting a symplex by certain pencils of hyperplanes through a point have volumes proportional to the Eulerian numbers a(n, k). R. Ehrenborg, M. Readdy and E. Steingrimsson [148] give a combinatorial interpretation for the mixed volumes of two adjacent slices from the unit cube in terms of a reﬁnement of the Eulerian numbers. M. Dash, L. Samantaray and R. Panda

~~568 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

[127] study the interpolation of band limited signals using Eulerian ﬁlters, and estab- lish a relation of different derivatives of a sigmoid function to the standard Eulerian numbers. There exist many identities involving Eulerian numbers. The following show a connection with the Stirling numbers of the second kind, and with Bernoulli numbers. The identity n n 2 j−1 A(n, j) = j!S(n, j), (205) j=1 j=1 has been proposed in [483], and generalized by R. Tauraso (see [483]) to:

~~∞ n xn+1− j n x j A(n, j) = j!S(n, j) = j n x j (206) ( − )n+1 ( − ) j+1 j=1 1 x j=1 1 x j=1~~

~~∞ 1 for |x| < 1. For x = , (206) gives (205) with a common value equal to j n/2 j . 2 j=1 The identity~~

n n+1 n+1 2 (2 − 1)B + (− ) j a(n, j) = n 1 (n ≥ )() 1 + 1 207 j=1 n 1 is well known. Foraproof, see J. Stopple [429] (who states that (207) may have been known to Euler). Since ζ(−n) =−Bn+1/(n + 1), (207) can be written also in terms of ζ(−n). There are a lot of generalizations or extensions of the Eulerian numbers. For example, in 1954 L. Carlitz [82] deﬁned the q-Eulerian numbers Bn,k(q) and polynomials Bn(x) by

~~k Bn,k(q) = [k + 1]Bn−1,k(q) + q [n − k]q Bn−1,k−1(q), (208) with B0,k(q) = 1ifk = 0; 0, otherwise, and~~

~~n k Bn(x) = Bn,k(q)x k=0~~

Here q ≥ 1and[k + 1] = [k + 1]q is the standard q-notation. For combinatorial and statistical properties, see e.g. L. Carlitz [81] and M. Skandera [404]. Another q-Eulerian polynomial An(x, q) has been introduced in 1971 by R. P. Stanley [417] (see also [418]). As usual, the q-ascending factorial is deﬁned by n−1 (u, q)0 = 1, (u, q)n = (1 − u)(1 − uq)...(1 − uq )(n ≥ 1). Let (u, q)∞ =

~~569 CHAPTER 5~~

∞ n n −1 (1−uq ),andput e(u, q) = u /(q, q)n = (u, q)∞ . Let desπ denote the num- n=0 n≥0 ber of descents of π of the permutation group Sn (i.e. the number of i ∈{1,...,n−1} with π(i)>π(i + 1)), and invπ the number of inversions of π (i.e. the number of pairs (i, j) with 1 ≤ i < j ≤ n such that π(i)>π(j)). Then An(x, q) is deﬁned by 1+desπ invπ An(x, q) = x q π∈Sn Then the generating functions of these polynomials are given by (see [418]) tn 1 − x A (x, q) = ,( ) n ( , ) − (( − ) , ) 208 n≥0 q q n 1 xe 1 x t q so that this is equivalent to the recurrence relation

n−1 n−1 n n−1−i An(x, q) = x(1 − x) + Ai (x, q)x(1 − x) i=1 i 1+desπ We note that for q = 1, An(x, q) = An(x) = x ,sowegetthe recur- π∈Sn rence from (204) in this case. =− −( ) = ( , − ) For q 1 one obtains the signed Eulerian polynomial An x An x 1 , introduced by J. L. Loday [289] in connection with his study of the cyclic homology of commutative algebras. Loday conjectured that − ( ) = ( − ) − ( ), − ( ) = ( + ) − ( ) + ( − )( − ) ( ) A2n x 1 x A2n−1 x A2n+1 x 2n 1 xA2n x x 1 x A2n x or equivalently − ( ) = ( − )n ( ), − ( ) = ( − )n ( ), A2n x 1 x An x A2n+1 x 1 x An+1 x and this has been proved by J. Desarm´ enien´ and D. Foata [135]. The q-Eulerian polynomials Am,n(x, q)(m, n ≥ 0) with two indices were introduced by J. Desarm´ enien´ [134] via their generating function tn sm 1 − t A , (x, q) · = m n ( , ) − (( − ) , ) (1−t)s m≥0,n≥0 q q n m! 1 xe 1 x t q e

Then A0,n(x, q) = An(x, q), Am,0(x, q) = Am(x). He proved the following congruence property (see also [135] for a detailed proof): Let n and k be two positive integers and let n = ka + b (0 ≤ b ≤ k − 1) be the Euclidean division of n by k. Then

~~(k−1)a Am,ka+b(x, q) ≡ (1 − x) Am+a,b(x, q)(mod k(q)), (208 )~~

~~570 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~where k is the kth cyclotomic polynomial. For k = 1wethenhaveAa,1(x, q) = Aa+1,0(x, q) = Aa+1(x); for k = 2, b = 0, 1 one has~~

~~a A2a+b(x, q) = A0,2a+b ≡ (1 − x) Aa,b(x, q)(mod (q + 1)) ≡~~

a ≡ (1 − x) Aa+b(x)(mod (q + 1)), − − ( ) so Loday’s conjecture on A2n and A2n+1 is a particular case of 208 .Foran involution for signed Eulerian numbers, see M. Wachs [477]. Recall that these numbers A−(n, k) satisfy

~~A−(2n, k) = A−(2n − 1, k) − A−(2n − 1, k − 1),~~

A−(2n + 1, k) = kA−(2n, k) + (2n − k + 2)A−(2n, k − 1), implying analogs of the Worpitzky formulas 2n + i 2n − 1 + i A−(2n, k − i) = kn, A−(2n − 1, k − i) = kn i i i i F. Chung, P. Diaconis and R. Graham [108] have considered the generalized Eulerian numbers a(n; k, l) deﬁned by

~~a(n; k, l) = (l + 1)a(n − 1; k, l) + (l + 1)a(n − 1; k − 1, l + 1)+~~

+(n − k − l)a(n − 1; k − 1, l) + (k − l + 1)a(n − 1; k, l − 1)(209) where a(0; 0, 0) = 1anda(m; n, p) = 0ifm, n or p < 0. P. L. Butzer and M. Hauss [71] have introduced the Eulerian functions a(α, k), α ∈ R,which for α = n ∈ N are given by (199). The a(α, k) satisfy recursion formulae, they are monotone in k,and as functions of α, are arbitrarily often differentiable. A counterpart of the Worpitzky formula (203) is given, as well as connections to the fractional Stirling numbers S(α, k) are pointed out (see relation (98) of 5.2.). ForEulerian numbers of higher order, see J. F. Dillon and D. P. Roselle [145]. For ”Pseudo-Eulerian” numbers, see [100]. In [227] Hsu and Shiue, by using the Dickson-Stirling numbers S(n, k,α) (see (116) of 5.2.)have deﬁned the Dickson-Eulerian polynomial by

~~n k n−k An(x,α)= k!S(n, k,α)x (1 − x) , k=0 and the Dickson-Eulerian numbers by~~

~~n k An(x,α)= A(n, k,α)x k=0~~

~~571 CHAPTER 5~~

The numbers A(n, k,α)have many properties, e.g. n n − i A(n, k,α)= i (− )k−i S(n, i,α) ! 1 − i=0 n k They also have introduced the degenerate Eulerian numbers A(n, k,λ|θ), which satisfy e.g. Worpitzky type formulas. ˇ m Starting from the identity (202), Z. Sami [381] has introduced the numbers ak,n where n ≥ 1, j ≥ 0, m ∈{0, 1,...,n − 1},bydeﬁning k n m = (− ) j ( + − )n−m−1( − )m ( ) ak,n 1 k 1 j k j 210 j=0 j

0 = 0 = ( − , + ) n−1 = ( − , ) with the convention 0 1. Therefore ak,n A n 1 k 1 and Ak,n A n 1 k , m so the numbers ak,n can be considered as generalization of the Eulerian numbers A(n, k). m These numbers ak,n satisfy the recurrences m = ( + ) m + ( − − ) m ( ≤ ≤ − ) ak,n k 1 ak,n−1 n k 1 ak−1,n−1 0 m n 2 ( ) m = m−1 − m−1 + m−1 ( ≤ ≤ − ), 211 ak,n ak,n ak,n−1 ak−1,n−1 1 m n 1 where k ≥ 1 and n ≥ 2. ≥ m = Further, if k n, then ak,n 0 and 0, if 0 ≤ m ≤ n − 2 am = (212) n−1,n 1, if m = n − 1 and have the symmetric property m = n−m−1 ≥ ; , ∈{ ,..., − } ( ) ak,n an−k−1,n for all n 1 m k 0 n 1 213 The following identities are true:

~~n−1 m = ( , + )( ≥ , ≥ ), ak,n A n k 1 k 0 n 1 m=0~~

~~n−1 am = (n − 1)! (n ≥ 1, 0 ≤ m ≤ n − 1), k,n (214) k=0 n−1 m n−i − k m n i m 2 1 (− ) a = (− ) B − 1 k,n 2 1 − n i k=0 i=0 i n i (n ≥ 3, 1 ≤ m ≤ n − 2)~~

~~572 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~Let n−1 = i ( ≥ , ≥ ) Sk,n ak+i,n n 1 k 0 i=0~~

~~Then Sk,1 = 0ifk ≥ 1; = 1ifk = 0, and~~

~~n−1 k Sk,n = Sk,1+!n − A(r, j)(215) r=0 j=0~~

~~n−1 with !n = k!denoting the left-factorial function of Kurepa [266]. Applying (215) k=0 for k = 0andk = 1, one gets~~

~~n−1 n−2 i = , i = − ( ) ai,n !n ai+1,n !n n 216 i=0 i=0~~

~~From (216) it follows that for p ≥ 3 prime,~~

~~p−1 !p ≡ (i + 1)p−i−1i i (mod p), i=0 (217) p−2 !p = (i + 1)p−i i i−1 (mod p) i=1~~

Then Kurepa’s conjecture that !p ≡ 0 (mod p) ∀ p ≥ 3isequivalent to the following: p−2 (i + 1)p−i i i−1 ≡ 0 (mod p) ∀ p ≥ 3 (218) i=1 An identity as the second one of (214) is valid for the classical Eulerian numbers, too (due to Worpitzky [487])

~~n A(n, k) = n! (219) k=1~~

Now, N. Robbins [368] proved the followinf congruence for A(n, k):Ifp is prime and m ≥ 1, 1 ≤ k ≤ pm − 1, then 0 (mod p), if k ≡ 0 (mod p) A(pm − 1, k) ≡ (220) 1 (mod p), if k ≡ 0 (mod p)

~~573 CHAPTER 5~~

So A(p − 1, k) ≡ 1 (mod p) for 1 ≤ k ≤ p − 1, and by (219), (p − 1)! = p−1 A(p − 1, k) ≡ p − 1 ≡−1 (mod p).Therefore (220) and (219) imply the k=1 Wilson theorem: (p − 1)! ≡−1 (mod p). For other arithmetic properties of Eulerian numbers, see e.g. L. Carlitz and J. Riordan [98], R. A. Saenz´ Casa and J. E. Nymann [374], or K. Knopfmacher and N. Robbins [256]. Forexample, in [374] one can ﬁnd that for all primes p and all 1 ≤ k ≤ p one has A(p, k) ≡ 1 (mod p)(220 )

~~Let P(n, k) = k!S(n, k) (see Part D. of section 4 of 5.2.,e.g. relation (194)). Then (220’) is equivalent to the following: 1 (mod p) if k = 1, P(p, k) ≡ (220 ) 0 (mod p) if 2 ≤ k ≤ p~~

~~These congruences follow in fact from the inversion formulas~~

~~− k 1 n − k + i P(n, k) = A(nk − i), i=0 i~~

~~− k 1 n − k + i A(n, k) = (−1)i P(n, k − i) i=0 i p − 1 combined with the known relation (−1)k−1 ≡ 1 (mod p) for 1 ≤ k ≤ p. k − 1~~

4 Estimates and inequalities Estimates and inequalities for Bernoulli or Euler (and Eulerian) numbers have a central importance in many areas connected with related ﬁelds. ∞ 1 A. The important property (7) gives the values of ζ(2n) = via B .A 2k 2n k=1 n similar property for the Euler numbers is

~~∞ (−1)k π 2n+1 = (−1)n E (221) ( + )2n+1 2n ( ) 2n+2 k=0 2k 1 2n !2~~

~~574 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~Relations (7) and (221) yield various inequalities for the Bernoulli or Euler numbers. For example, Ch. Jordan [237] showed that~~

2(2n)! 2(2n)! π 2 < |B |≤ · (222) (2π)2n 2n (2π)2n 6 and 24(2n + 1)(2n + 2) B + (2n + 1)(2n + 2) < 2n 2 < (223) 4 2 (2π) B2n (2π) Various authors have given improvement to these inequalities, e.g. D. J. Leeming [273], A. Laforgia [268], D’Aniello Ciro [110], H. Alzer [20]. For example, in [110] one can ﬁnd the inequalities 2(2n)! 1 2(2n)! 1 · < |B | < · (224) (2π)2n 1 − 2−2n 2n (2π)2n 1 − 21−2n We note that the upper bound can be found also in Abramovicz-Stegun [1] from 1965 (where the lower bound of (222) is included, too). Leeming proved that √ n 2n √ n 2n 4 πn < |B | < 5 πn (n ≥ 2), (224 ) πe 2n πe and Alzer obtained the sharp bounds 2(2n)! 1 2(2n)! 1 · < |B |≤ · (n ≥ 1)(224 ) (2π)2n 1 − 2α−2n 2n (2π)2n 1 − 2β−2n where the best possible constants are α = 0 (i.e. the lower bound in (224)), and log(1 − 6/π 2) β = 2 + = 0.6491 ... log 2 It can be shown that (224”) sharpens also the above bounds (224’) given by Leeming. By remarking that π 2(1 − 21−2n)4 ≥ 1 for all n ≥ 2, the upper bound of (224) and the lower bound of (222) imply (4n)!(4n − 2)! B B − ≤ 4n 4n 2 (n ≥ 2), (225) ( )4 4 2n! B2n which was a problem due to L. I. Nicolaescu [326]. By using (224), the left side of (223) can be improved too: (2n + 1)(2n + 2) B + (2n + 1)(2n + 2) x < 2n 2 < ,(226) n 2 2 (2π) B2n (π)

~~575 CHAPTER 5~~

~~2n+2 2n+2 where xn = (2 − 8)/(2 − 1).By(224) and (226) one obtains~~

2(2n)! |B |∼ (227) 2n (2π)2n and 2 B + n 2n 2 ∼ as n →∞ (228) 2 B2n π For estimates of type B2n 2 −2n − = O((4π − ε) ) (2n)! (2π)2n as n →∞, (ε > 0),see A. M. Odlyzko [339]. Similarly, (221) implies 4n+1(2n)! |E |∼ (229) 2n π 2n+1 and 2 E2n+2 4 2 ∼ n as n →∞ (230) E2n π and similar inequalities to (222)-(226) for |E2n| can be deduced. Remark that for |B2n| a similar formula to (221) holds true:

∞ (−1)k−1 |B | = (1 − 21−2n)ζ(2n) = (22n−1 − 1) 2n π 2n,(231) 2n ( ) k=1 k 2n ! and depending on the number of summands one can get various lower bounds. By using the Euler-Maclaurin summation formula (see e.g. [338]) one can deduce the asymptotic expansions:

~~π 2 N−1 ∞ − 1 ∼ 1 + B2m = 2 2 2m+1 6 k=1 k 2N m=0 N~~

~~1 1 1 1 1 = + + − + − ... (232) N 2N 2 6N 3 30N 5 42N 7 and~~

~~/ ∞ π [N 2] (−1)k−1 E 1 1 5 61 − 2 ∼ 2m = − + − + ... (233) − 2m+1 3 5 7 2 k=1 2k 1 m=0 N N N N N~~

~~576 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~The above inﬁnite series are clearly divergent, the correct interpretation of their asymptotics is~~

∞ B K B (2K + 1)! 2m = 2m + O , 2m+1 2n+1 ( π )2K +1 m=1 N m=1 N 2 N (234) ∞ E K E (2K + 1)! 2m = 2m + O 2m 2m (π )2K +1 m=1 N m=1 N N where K is a ﬁxed positive integer. By using the Boole summation formula (see e.g. [237], [338]), the following more precise result is valid (see J. M. Borwein, P. B. Borwein and K. Dilcher [59]): For positive integers n and M we have

∞ (−1)k M 2E 4 = (−1)n 2k + R (M), (235) + ( )2k+1 1 k=n 2k 1 k=0 2n where 2|E | |R (M)|≤ 2M 1 (2n)2M+1 By similar methods, for the tangent numbers given by (9) (where, in fact T (0) = n n 1, 2k T (n − k) + T (n) = 0forn ≥ 1) one can deduce: k=0 k " ∞ (−1)k+1 1 M T (2k − 1) = (−1)n+1 + + R (M), (236) ( )2k 2 k=n+1 k 2n k=1 2n with |E | |R (M)|≤ 2M 2 (2n)2M+1 Forageneralization of the Euler-MacLaurin and Boole summation formula, see Berndt [45]. B. The sequence of Eulerian numbers (A(n, k)) is unimodal in k for any ﬁxed n:

A(n, k − 1)A(n, k + 1) ≤ (A(n, k))2 (0 ≤ k ≤ n), (237) see L. Comtet [116] and V. Gasharov [177]. For a new (combinatorial) proof of this result, see M. Bona´ and E. Ehrenborg [54], where the log-concavity of (R(n, k))k is proved, too (R(n, k) = number of permutations of {1, 2,...,n} with k runs). More generally, the sequence (A(n, k)),0≤ k ≤ n,isaPolya´ frequency (PF - in short) sequence,inthe following sense:

~~577 CHAPTER 5~~

An infinite sequence a0, a1,... of nonnegative numbers is called a Polya´ frequency sequence, if the matrix (ai− j )i, j≥0 is a totally positive (TP - for short) matrix, i.e. if all minors of this matrix have nonnegative determinants (suppose ak = 0ifk < 0). A finite sequence a0,...,an is PF-sequence if the infinite sequence a0, a1,...,an, 0, 0, 0,... is PF. (Thus, a PF sequence is necessarily log-concave), The following theorem due to M. Aissen, I. J. Schoenberg and A. Whitney [16] gives a characterization of finite PF-sequences (see also Curry and Schoenberg [125]): Afinite sequence a0,...,an of nonnegative numbers is PF if and only if its gen- n i erating function (polynomial) ai x has only real zeros. (238) i=0 The fact that (A(n, k))k is PF, can be found in [116]. L. H. Harper [213] proved this fact for the Stirling numbers of the second kind (S(n, k))k, while L. A. Szekely´ [435] for his holiday numbers. J. C. Ahuja and E. A. Enneking [14] proved the PF- property of a sequence (Lm(n, k))k associated with the Lah numbers. Recently Y. Wang and Y.-N. Yeh [478] have studied a general sequence (X(n, k)) with nonnegative terms satisfying the recurrence

~~X(n, k) = (rn + sk + t)X(n − 1, k − 1) + (an + kb + c)X(n − 1, k)~~

(n ≥ 1, 0 ≤ k ≤ n) where X(n, k) = 0 unless 0 ≤ k ≤ n. They have proved that if the (given) real numbers r, s, t, a, b, c satisfy the following conditions: rb ≥ as,(r + s + t)b ≥ (a + c)s, then for each n ≥ 0, (X(n, k))0≤k≤n is a PF-sequence. (239) For PF-sequences, see also J. Pitman [350]. L. Lesieur and J. L. Nicolas [285] have proved the inequality

~~− + n − k n 2k 2 A(n, k − 1)< A(n, k)(n ≥ 2k − 1, k ≥ 2)(240) n − k + 2 which enabled then to show that the sequence~~

~~(M2n/(2n)!)n≥1~~

~~is decreasing, where Mn = max A(n, k).(241) 1≤k≤n~~

~~578 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

Further, by using Cauchy’s integral formula and the Laplace method, they have deduced M − 3 3 2n 1 ≤ 1 − (n ≥ 3), (2n − 1)! πn 40n M2n−1 3 3 13 ( ) ≥ 1 − − (n ≥ 1), 242 (2n − 1)! πn 40n 4480n2

M + M M + 2n 5 ≤ 2n ≤ 2n 3 (n ≥ 1) (2n + 5)! (2n)! (2n + 3)! See also [286]. For the Eulerian numbers a(n; k, l) given by (209), Chung, Diaconis and Graham [108] prove: − 4 l a(n; k, l)2−k = O(n!cn)(243) k,l 3 where 0 < c < 0.68528 ... The asymptotic normality of Eulerian numbers have been studied e.g. in L. Carlitz, D. C. Kurtz, R. Scoville and O. P. Stackelberg [79], and recently in L. H. Y. Chen, T. N. T. Goodman and S. L. Lee [105]. In [105] the following general theorem is proved: Let r , ( j = 1, 2,...,n) be complex numbers lying in the sector π n j | arg z|≤ , and deﬁne the numbers a , (n ≥ 1, k = 0, 1,...,n)by 3 n k n n k an,k z = (z + rn, j )/(1 + rn, j ) k=0 j=1

~~Assume that the numbers rn, j are bounded away from −1, the the coefﬁcients an,k are real, and that~~

n σ 2 = /( + )2 →∞ →∞ n rn, j 1 rn, j as n j=1 Then k − µ σ − n = (σ −1/2), ( ) max nan,k G O n 244 k=0,n σn n k and if an,k z is reciprocal, then k=0 k − µ σ − n = (σ −2/3)() max nan,k G O n 245 k=0,n σn

~~579 CHAPTER 5~~

1 −x2/2 Here G(x) = √ e , µn = µ(mn), where mn (n ≥ 1) denote the discrete 2π ({ }) = ( = , ,..., ) measures deﬁned by mn k an,k k 0 1 n . √ When an,k = A(n, k) (i.e. Eulerian numbers), then σn = π(n + 1)/6, and we get from (245) the order of convergence O(n−1/3),which improves O(n−1/4) of [79]. In 1965 L. Carlitz [78] considered the problem of expanding Gm(λ) = zn nn−m in terms of λ = nn−1znn!. He proved that n≥1 n! n≥1

~~n λ m n+m z k G− (n) = n = E(m, k)λ (246) m ( − λ)2m+1 n≥1 n! 1 k=0 The coefﬁcients E(m, k) satisfy the recurrence~~

~~E(m, k) = (k + 1)E(m − 1, k) + (2m − k − 1)E(m − 1, k − 1)~~

For combinatorial interpretations, see J. Riordan [366] and I. Gessel and R. Stanley [188]. In the book [200] of Graham, Knuth and Patashnik, the notation m E(m, k) = is applied, where these coefﬁcient are called as the ”second k order Eulerian numbers”. See also L. M. Smiley [412]. There is an interesting connection between E(n, k) and the associated Stirling numbers of the second kind b(n, k) (see section 4. of 5.2.) which count the ways in which n distinct objects may be partitioned into exactly k true heaps (a true heap has more than one object). One has the identities

n n E(n, k)xk = b(n + k, k)xk−1(1 − x)n−k, and k=0 k=1 (247) n n E(n, k)(1 + x)n−k−1xk = b(n + k, k)xk−1, k=0 k=1 see L. M. Smiley [413]. We note that here Smiley uses the notation E(n, k) =n, k and b(n, k) ={{n, k}}. C. As we have seen (see e.g. in connection with (112)) the higher order Bernoulli µ( ) polynomials can be generalized to ”complex orders”, i.e. by deﬁning Bn x for all x,µ∈ C by ∞ µ tn t Bµ(z) = ezt , |t| < 2π n t − n=0 n! e 1

~~For µ = 1wehave the Bernoulli polynomials Bn(x).Forx = 0weget the µ µ µ = Bernoulli numbers Bn of order , which for 1give the classical Bernoulli~~

~~580 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

numbers Bn.Aswehaveseen also, the classical Stirling numbers of the first and second kind are in fact higher order Bernoulli numbers, since − n 1 n n −m s(n, m) = B − , S(n, m) = B − m − 1 n m m n m In 1961 N, E. Norlund¨ [337] has shown that if ρ and x are independent of n, then − Bn+ρ+1(z) (−1)n(log n)ρ m1 ρ (−1)k n ∼ A (z) + O(log n)−m (248) z ( )k k n! n k=0 k log n as n →∞. The coefficients of Ak(z) are derivatives of the reciprocal gamma function: dk 1 A (z) = k dzk (1 − z) When ρ ≥ 0isaninteger, then (248) reduces to a finite number of terms (because the binomial coefficient vanishes when k >ρ). In particular, when ρ = 0, one has n+1( ) = ( − )( − )...( − ) Bn z z 1 z 2 z n .Therefore ∞ Bn+1(z) (n + 1 − z) B1−z (−1)k n = (−1)n ∼ k · , ( − )( + ) ( − − ) z+k n! 1 z n 1 k=0 k! 1 z k n which is a well-known result for the ratio of two gamma functions. Forfixedvalues of n Norlund¨ gives the expansion 1 2cosπ 2z + µ − n Bµ(z) 2 1 n = nµ−1 1 + O (249) n! (µ)(2π)n n µ Since Bn can be represented as a Cauchy type integral µ n! t dt Bµ(z) = ezt ,(250) n t n+1 2πi C e − 1 t where the contour C is a circle of radius less than 2π around the origin, N. M. Temme µ( ) [442] recently has generalized Bn x for complex n, via µ (ν + 1) t dt Bµ(z) = ezt (251) ν t ν+1 2πi C e − 1 t with Re z > 0. Because of the algebraic singularity of tν+1 at the origin, we assume that the contour of integration runs like a Hankel integral from −∞ around the origin

581 CHAPTER 5 to −∞ (i.e. from −∞,argt =−π encircles the origin in positive direction, termi- nates at −∞, with arg t =+π). We assume that all zeros of et − 1(except t = 0) are not enclosed by the the contour, and initially, the many valued function tν is real for real values of ν and t > 0. By using analytic continuation, the Bernoulli polynomials B(z) (i.e. when µ = 1 in (251)) can be deﬁned in the sector | arg z| <π,forany ν ∈ C.Byusing (251) it follows that the two basic properties of Bernoulli polynomials remain valid: ν−1 Bν(z + 1) = Bν(z) + νz (ν ∈ C, | arg z| <π), d (252) Bν(z) = ν Bν− (z)(ν∈ C, | arg z| <π) dx 1 Generalizations of this type have origins in the classical papers by A. Jonquiere` [236] and P. Bohmer¨ [53]. See also P. L. Butzer, M. Hauser and M. Leclerc [68], [69] (who have overlooked the Jonquiere` and Bohmer¨ papers) for generalizations of Bernoulli and Euler polynomials, and related results. n n n−k The well known property Bn(z) = Bk z has now the following ana- k=0 k logue: ∞ ν ν−k Bν(z) ∼ Bk z as z →∞, | arg z| <π (253) k=0 k It follows that, when Re ν<0, then

Bν(z) → 0asz →∞in | arg z| <π (254) Further, the following Maclaurin series expansion is valid (see [442]): ∞ (− )k ν−1 1 z Bν(z) =−νz + (k + − ν)ζ(k + − ν) ,() (−ν) 1 1 255 k=0 k! where |z| < 1. This reduces to the finite polynomial expansion of Bn(z) when ν = n. The analytic continuation of Bν(z) from the half plane Re z > 0 into the left half of the complex plane also follows from the first equality of (252). Also in this way we cannot reach the negative z-axis. For the definition of Bν(x) for ν>0, remark that this depends on the way we approach the negative z-axis: from above or from below. Taking the average of the two values obtained so, we define (see [442])

~~∗ Bν(x + iy) + Bν(x − iy) Bν (x) = lim , x < 0 (256) y→0 2~~

~~Then 1 ∞ sin 2πkx − νπ ∗ 2 B (−x) = cos πνBν(x + 1) + 2(ν + 1) sin πν (257) ν ( π )ν k=1 2 k~~

~~582 STIRLING, BELL, BERNOULLI, EULER, AND EULERIAN NUMBERS~~

~~∗ for x > 0, Re ν>1; and Bν (x) satisﬁes the following difference property: ν−1 ∗ ∗ νx , if x ≥ 0 B (x + 1) − B (x) = ν ν −ν|x|ν−1 cos πν, if x < 0, Re ν>1~~

For another deﬁnition of Bν(x) for x < 0, see Butzer et al. [68]. µ For asymptotic results on Bν , via the saddle point method, see Temme [442]. The method is based of Stirling asymptotics from [439]. We quote only the following result. Assume that ν is large and positive, and the µ is a real parameter. Let

~~φ(t) = (µ − ν)log t − µ log(et − 1).~~

Then µ (ν + 1) et dt Bµ = · eφ(t) , ν t µ − ν 2πi C e − 1 d so to calculate the saddle points, i.e. the solutions of φ(t) = 0, we have to solve dt the equation µ − e−t = λt λ = ( ) 1 µ − ν 258 When λ<0, the equation has an inﬁnite number of complex solutions, which occur in complex conjugate pairs, and one pair {t+, t−} can be used for the saddle point method. Locally at t = t+ or t = t− we can approximate φ(t) up to the quadratic term of its Maclaurin expansion, and we obtain the asymptotic result: φ(t−) φ(t+) µ (ν + 1) e e Bν ∼ √ √ − √ as ν →∞ (259) 2πi t− −φ (t−) t+ −φ (t+)

~~The uniform asymptotics of Bν(z) for large values of z is based on earlier work on asymptotic expansions of Laplace integrals, see F. W. J. Olver [343], N. M. Temme [440], [441], R. Wong [486].~~

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~~618 Index~~

(A, r)-powerful, 114 k-Eulerian posets, 148 (m, k)-perfect, 41, 42 k-free part, 134 (α, β)-multiamicable numbers, 71 k-full, 236 3x + 1 problem, 190 k-full part, 134 A-analogue of Smith’s k-harmonic number, 44 determinant, 266 k-hyperperfect numbers, 50 A-convolution, 112, 278 k-Mobius¨ function of a poset, 147 A-multiplicative, 114 k-multiperfect numbers, 30 A-rational function, 114 k-perfect number, 32 Ak-convolutions, 113 k-perfect numbers, 33 B-convolution, 116 k-reduced residue system, 276 B-splines, 471 k-Smith number, 383 b-additive, 404 k-th power of an arithmetic e-multiperfect, 52 function, 108 e-perfect numbers, 52 k-unitary perfect number, 46 e-superperfect numbers, 53 kth cyclotomic polynomial, 561 f -expansions, 432 kth iterate of f , 196 f -generated number, 386 l p norm (p ≥ 1) of matrices, 274 f -iterative sequence, 427 M-void analogue of the Euler f -self-number, 386 totient, 290 G-ary digital expansion, 414 m-bi-unitary perfect numbers, 57 G-convolution, 121 m-bi-unitary superperfect numbers, 58 G-multiplicative functions, 121 m-integral, 539 jth iterate of the Carmichael m-multisuperperfect numbers, 55 function, 200 m-perfect numbers, 55 K convolution of incidence m-superperfect numbers, 39, 56 functions, 143 m-unitary (k, l)-perfect K -convolution, 114, 115 number, 57 K -iterate of an arithmetical m − e-perfect numbers, 58 function, 115 m − e-superperfect numbers, 58 k-ary divisor, 136 n-dimensional symplex, 568

619 INDEX n-permutations with η-colored q-Euler function, 286 head, 480 q-Euler numbers, 561 n-th prime pn, 180 q-Euler polynomials, 563 nth prime, 234 q-Eulerian numbers, 569 nth symmetric mean, 471 q-Eulerian polynomials, 570 p-adic L function, 566 q-exponential polynomial, 483 p-adic analysis, 541 q-generated number, 384 p-adic expansion, 498 q-Hurwitz ζ-function, 565 p-adic gamma function, 566 q-log- function, 563 p-adic integers, 550 q-multinomial coefficients, 506 p-adic integrals, 566 q-multiplicative, 401, 405, 413 p-adic log-gamma functions, 566 q-Niven number, 382 ζ p-adic mean value theorem, 546 q-Riemann -function, 562 p-adic method, 494 q-Stirling numbers, 481, 504 p-group, 155 q-tangent numbers, 559 p-local integers, 553 q-trigonometric function, 561 − p-Stirling numbers, 472 q L-series, 563 p, q-log-concavity, 515 r-Stirling numbers, 469 p, q-Stirling numbers, 484 S-perfect numbers, 55 t-chain, 370 Q-additive function, 389 Z-perfect numbers, 55 Q-multiplicative, 406 α-adic representation, 419 q-additive, 387, 401 γ -perfect numbers, 54 q-additive functions, 388 ψ-multiperfect numbers, 41 q-analog of log-concavity, 513 ψ-superperfect numbers, 40 q-analogs of Euler numbers, 559 ψ ◦ σ-perfect numbers, 40 q-analogue of a positive integer, 481 σ ◦ ψ-perfect numbers, 40 q-analogue of Bell numbers, 486 ϕ-amicable pairs, 71 q-analogue of Jordan’s totient, 286 ϕ-convergence, 187 q-analogue of power series, 486 ϕ-partitions, 223 q-analogue of the p-adic measure, 566 R-multiplicative, 407 q-analogue of the classical Euler R-system, 407 totient, 286 U-Stirling numbers, 485 q-analogue of the classical Ramanujan ”breeding” method by Borho and sum, 286 Hoffman, 63 q-ascending factorial, 569 q-Bernoulli numbers, 483, 561 Abelian minimal normal subgroup, q-Bernoulli polynomials, 562 155 q-binomial coefficient, 471 abelian number fields, 553 q-binomial coefficients, 480, 506 abstract prime number theorems, 154

620 INDEX abundant numbers, 17, 43 analytic continuation, 582 action of the cyclic group, 490 and Mellin-Perron summation adding prime totatives, 248 formula, 423 adding composite totatives, 248 aperiodic elements, 490 additive analogue of Euler’s totient, Apostol function, 129 291 applet, which computes the values additive analogue of the σ, 292 ϕ(n) and λ(n), 193 additive arithmetic function, 366 applications of Mobius¨ inversion adjacent slices, 568 formulae, 156 Airy asymptotic, 524 applications of the Mobius¨ functions, algebraic function fields, 161 139 algebraic independence of real valued Approximation theory, 469 arithmetic functions, 185 arithmetic process, 343 algebraic integers, 293, 421 arithmetic progression, 221 algebraic number field, 159, 263, 419 arithmetic progressions, 136, 151, algebraic number fields, 193 220, 229, 254 algebraic number theory, 160 arithmetical products, 285 algebraic numbers, 553 arithmetical semigroup, 149 Algebraic topology, 162 arithmetical semigroups, 292, 331 algebraic topology, 499 arithmetical triangles, 502 algorithms, 36, 419 ascents, 567 aliquot sequence, 72, 74 associated Stirling numbers, 502, 580 all-Niven, 383 asymptotic analysis, 511 Alladi’s totient, 282 asymptotic behavior of the Stirling almost all sums, 340, 342 numbers, 516 almost amicable pairs, 69 asymptotic density, 363 almost perfect numbers, 37, 38 asymptotic development of the almost superperfect numbers, 41 Stirling numbers of the first amicable k-tuples, 70 kind, 515 amicable numbers, 36, 60, 63 asymptotic expansions of Stirling amicable numbers in special numbers of the second kind, sequences, 67 520 amicable triples, 69 asymptotic normality, 416 analogue of Adams’ theorem, 557 asymptotic normality of Eulerian analogue of Euler’s formula, 62 numbers, 579 analogue of Lehmer’s problem, 213, asymptotic prime divisor, 341 214 asymptotic sign fluctuations, 517 analogue of Smith’s determinant, 266 asymptotically Poisson distributed, analogue of Thabit’s formula, 62 518

621 INDEX asymptotically unit exponential binary recurrence sequence, 240 variables, 341 binomial coefﬁcient, 49, 107, 126, 371 automorphic numbers, 431 binomial coefﬁcients, 243, 460, 474 average number of blocks, 512 binomial convolution of sequences, 117 Baker’s method, 418, 430 binomial measure, 375 Barrucand numbers, 494 biological analogies, 17 base q palindromes, 430 bipartite graphs, 348 basis, 481 birational transformation, 192 Bassily-Katai´ theorem, 403 block-additive, 410 Bauer’s theorems, 245 block-multiplicative, 408 BDE method, 61, 63 Bohr almost-periodic function, 136 Bell number, 461 Bohr-Fourier spectrum, 413 Bell numbers, 493 Bolyai-Renyi´ expansions, 432 Bell polynomials, 484, 495 Bombieri’s sieve, 227 Bell sequence, 512 Boole summation formula, 577 Benford law, 430 Boolean lattice, 513 Bernoulli and Euler polynomials, Borel σ-algebra, 344 529, 582 bounded below set, 148 Bernoulli and Stirling numbers, bounded variation, 103 259, 467 Brauer-Rademacher identity, 144, 180 Bernoulli number, 461 britannic number system, 407 Bernoulli numbers, 243, 244, 495, Brownian motion, 344 500, 518, 525 Brun’s sieve, 235 Bernoulli numbers of fractional order, Brun-Titchmarsh theorem, 216, 227 474 Buchstab function, 357 Bernoulli numbers of higher order, Burnside basis theorem, 293 500 Bernoulli numbers of order k, 468 C++ programming language, 217 beta-expansions, 432 cancellation property, 120 betrothed numbers, 68 canonical basis, 393 Beurling’s generalized primes, 207 canonical number system, 418 bi-unitary amicable numbers, 67 canonical number systems, 404 bi-unitary convolution, 111 Cantor representations, 412 bi-unitary divisor, 46 Carlitz congruence, 552 bi-unitary multiply perfect numbers, Carmichael conjecture, 228 47 Carmichael’s λ function, 193 bi-unitary perfect numbers, 47 Carmichael’s theorem, 190 bi-unitary quasiperfect, 47 Cashwell-Everett totient, 280 bilinearly recurrent orthogonality, 476 Catalan’s conjecture, 73

~~622 INDEX~~

Cauchy product, 117, 122, 128 composition of some arithmetic Cauchy’s integral formula, 579 functions, 39 central factorial numbers, 479 computational evidence, 42 central limit theorem, 344, 402, 423 computer algebra, 486 chain, 72, 369 conductor, 553 chains, 141, 144 congruence of Kummer, 541 change of the sum of digits by congruence of Voronoi, 541 multiplication, 378 congruence properties, 477 character group, 151 congruence properties of σ (m)(n),42 characteristic equation, 240 congruence properties of Euler’s characteristic polynomial, 414 totient, 201 characters of ﬁnite abelian groups, 151 congruence property, 188 Chebysheff’s theorem, 207 congruences for Genocchi numbers, circle of convergence, 415 541 circulant matrix, 183 congruences for higher order circular permutations, 192 Bernoulli numbers, 549 classical Mobius¨ function, 99, 128, congruences for Stirling numbers, 488 135 conjecture, 42, 47, 51–53, 59, 195, code, 179, 396 229, 235, 241, 348, 382, 400, coefﬁcients of cyclotomic 494, 499, 556, 558, 573 polynomials, 256 conjecture of Sierpinski, 229 Cohen totient ϕS, 290 conjectured, 46, 352 combinatorial and statistical conjectures, 199, 216, 237, 557 applications, 480 conjugacy classes, 157 combinatorial applications, 107, 477 conjugate Bernoulli numbers, 538 combinatorics on words, 390 conjugate Bernoulli polynomials, 538 common generalization of Liouville’s consecutive divisors, 342 function and the Mobius¨ consecutive prime factors, 337 function, 137 constants, 181 commutative group, 106 continued fractions, 432, 537, 561 commutative ring, 110, 121 continued powers and roots, 432 commutative semigroup, 117, 118 convex polytope, 559 commutator subgroup, 158 convolution of incidence functions, complete asymptotic expansion, 520 139 complete symmetric functions, 471 convolution on the set of sequences, complex bases, 417 537 complex variables, 381 convolutions with unbounded unity, composition of ϕ and other functions, 128 234 core, 208

623 INDEX core-reduced residue system, 290 derangements and cycle indicators, core-reduced totient of J. Sandor´ and 507 R. Sivaramakrishnan, 290 derived k-cycle, 337 cosets, 151 derived sequence, 337 counterexample to Carmichael’s descents, 570 conjecture, 228 determinant, 269 Criptography, 15 determinants, 265 cross-convolution, 113 determinants involving q-Stirling cumulative distribution function, 397 numbers, 483 cycle, 72, 427 Dibag’s von Staudt-Clausen theorem, cycles, 73, 75, 183, 462, 479, 491 553 cyclic homology, 570 Dickman function, 359 cyclotomic Bernoulli numbers, 539 Dickson’s conjecture, 221 cyclotomic field, 539 Dickson’s theorem, 30 cyclotomic fields, 556 Dickson-Eulerian numbers, 571 cyclotomic polynomial, 27, 138, 139, Dickson-Stirling numbers, 477 203, 251, 256, 506, 571 difference operator, 123 cyclotomic polynomials, 24, 261 differential equation, 567 differential equations, 469 Daykin’s Mobius¨ inversion formula, differential geometry, 390 110 Dedekind arithmetic function, 39 digits of Fibonacci numbers, 430 Dedekind arithmetical function, 231, Diophantine approximations, 263 284 Diophantine equations, 28, 61 Dedekind domains, 191 Diophantine Equations, 107 Dedekind zeta-function, 293 diophantine equations, 219 deficient numbers, 17, 43 direct factor set, 135 degenerate Eulerian numbers, 572 directed graph, 183 degenerate weighted Stirling numbers, Dirichlet convolution, 112, 117, 127 476 Dirichlet density, 152 degree of resolution, 367 Dirichlet inverse, 120 dense, 331 Dirichlet inversion, 123 density, 221 Dirichlet product, 106, 109, 119, 132 density of e-perfect numbers, 52 Dirichlet series, 100, 131, 243, 423 density of k-perfect numbers, 33 Dirichlet set, 206 density of amicable pairs, 65 Dirichlet’s theorem on arithmetical density of friendly pairs, 70 progressions, 210 density of harmonic numbers, 44 discrete analog of integration by parts, density of multiamicable numbers, 71 145 density of odd perfect numbers, 30 discriminator, 211

624 INDEX distance geometry, 486 Euclidean norm of a matrix, 274 distinct divisors, 347 Euler factor, 23, 30 distribution (mod 1) on divisors, 366 Euler functions of ﬁnite groups, 293 distribution function, 242, 345, 351, Euler gamma function, 563 353, 402, 524 Euler l.c.m. sequence, 211 distribution functions, 355 Euler maximal function, 210 distribution in residue classes, 206 Euler minimum function, 210 distribution of Bernoulli numbers Euler numbers, 527 modulo primes, 556 Euler quotients, 538 distribution of primes, 211 Euler totient, 40, 496 distribution of totatives, 246 Euler totient of matrices, 278 divided differences, 469 Euler’s divisibility theorem, 188, 538 divisibility algorithm, 191 Euler’s function, 22 divisor density, 366 Euler’s gamma function, 523 divisorial graph, 369 Euler’s methods, 61 divisors in residue classes, 363 Euler’s theorem, 24 divisors of factorials, 354 Euler’s totient, 67, 71, 106, 134, 137, Dobinski’s formula, 461 179, 184, 497, 544, 555 double series, 101 Euler’s totient function, 269, 272 double-factorial, 483 Euler-MacLaurin summation formula, dynamical system, 367 547 Egyptean fractions, 16 Euler-Maclaurin summation formula, eigenspace, 394 576 eigenvalue, 393, 409 Eulerian functions, 571 eigenvectors, 393 Eulerian numbers, 567 eingenvalues, 393 Eulerian numbers of higher order, 571 Eisenstein series, 544 Eulerian polynomials, 477, 568 elementary Abelian, 156 Eulerian posets, 146 elementary abelian, 159 even functions (mod n), 184 elementary symmetric function, 510 even perfect number, 19, 21, 22, 32, 54 epimorphisms, 158 even perfect numbers, 20, 22, 23, 44, equidistributed (mod 1), 366 59 Erdos-Kac¨ theorem, 388 even-odd amicable pair, 65 ( ) Erdos-Tur¨ an-Koksma´ inequality, 405 exponential analogue µ e of the Erdos-Wintner¨ theorem, 401 Mobius¨ function, 118 ergodic theory, 412 exponential Bell polynomials, 477 Euclid’s ”Elements”, 16 exponential convolution, 117 Euclid-Euler theorem, 21 exponential cycles, 75 Euclidean division, 570 exponential divisor, 51

625 INDEX exponential inverse, 118 Fibonacci sequence, 22 exponential operator, 108 Fibonacci, or Lucas numbers, 204 exponential sum of digit counting finite p-group, 293 function, 379 finite alphabet, 161, 374 exponential sums, 261 finite automata, 390 exponential totient function, 119, 288 finite field, 192, 286 exponentially A-multiplicative finite group, 154, 490 functions, 119 finite permutation groups, 193 exponentially multiplicative finite poset, 154 functions, 118 finite recurrences, 414 extended Nagell totient, 132 flag, 481 extension of Nagell’s (and Alder’s) formal languages, 161 totient, 279 formal Laurent series, 488 formal power series, 162, 426 factor-closed set, 267 Fourier analysis, 415 factorable function, 143 Fourier coefficients, 184, 544 factorable functions, 127 Fourier expansions, 531 factorial base, 386 Fourier transforms, 103 factorial numbers, 478 fractal dimension, 367 factorial ring, 106 fractional parts of the Bernoulli factorization methods of integers, 256 numbers, 540 factorization of Bernoulli numerators, Frattini chief factor, 155 556 free abelian group, 151 falling factorial, 478 frequency of a block, 421 falling factorial power, 460 friendly pair, 70 Fermat number, 223 Frobenius congruences, 544 Fermat numbers, 254 Frobenius’ classical theorem, 156 Fermat prime, 27, 191, 239, 527 function λ, 189 Fermat primes, 225, 236 function by Jacobsthal, 247 Fermat quotient, 548 functional limit theorems, 416 Fermat quotients, 530 fundamental domains, 421 Fermat’s ”little theorem”, 15 Fermat’s divisibility theorem, 179 Galois field, 481 Fermat’s little theorem, 488 Gauss norm, 547 Fermat’s quotient, 211 Gauss sum, 566 fermionic oscillators, 487 Gauss’ theorem, 193 Fibonacci and Lucas numbers, 182, Gaussian integer, 263 498 Gaussian integers, 32, 404 Fibonacci numbers, 398 Gaussian limit law, 399 Fibonacci or Lucas sequences, 239 Gaussian prime, 26, 263

~~626 INDEX~~

GCD matrices associated with generalized Thabit’s formulae, 61 arithmetical functions, 273 generalized totient function, 269 gcd-closed matrices, 270 generalized totient functions, 137 gcd-closed set, 270 generating function, 460, 461, 480, gcud-closed, 271 483, 505, 578 Gegenbauer’s theorem, 192 generating functions, 474, 476, 484, general theorem of log-convexity, 511 570 generalization of Euler’s theorem, Genocchi numbers, 532 191, 192 Genocchi numbers of order k, 533 generalization of Euler’s totient, 246 geometrical investigations of generalization of ﬁnite differences, congruence theorems, 189 475 Gill’s complexity class, 36 generalization of Mobius’¨ function, golden number, 486 115 Golomb-Guerin totient function, 288 generalization of the Mobius¨ function, Golubev totient functions, 289 137 good lattice points modulo composite generalizations of Euler’s totient, 291 numbers, 345 generalizations of Klee’s totient, 278 graceful graphs, 369 generalizations of Ramanujan’s sum, graded poset, 146 277 Gram’s determinant, 420 gravaritm, 201 generalizations of Ramanujan’s sum to matrices, 278 greatest common divisor, 263 greatest prime divisor of m, 255 generalizations of the Mobius¨ greatest prime factor, 356 function, 139 group of units of the ring Z , 249 generalized q-Bernoulli numbers, 562 n groups acting on functions, 490 generalized q-Euler numbers, 562 groups acting on graphs, 490 generalized arithmetical functions, Guerin analogue of the Mobius¨ 140 function, 122 generalized Bernoulli numbers, 553 generalized Eulerian numbers, 571 H-Hypothesis by Schinzel, 229 generalized harmonic numbers, 488 Hall π-subgroup, 158 generalized Liouville function, 114 Hankel contour, 523 generalized Mobius¨ function, 107, Hankel integral, 581 108, 135, 138 Hankel matrix, 483 generalized Mobius¨ functions, 134 Hardy-Littlewood circle method, 261 generalized Mobius¨ inversion, 104 Hardy-Littlewood conjecture, 219 generalized primes, 150 Harmonic analysis, 415 generalized Ramanujan sum, 114 harmonic number, 466 generalized Stirling numbers, 469 harmonic numbers, 43, 44, 472

~~627 INDEX~~

Haukkanen Mobius¨ function, 114 inequalities for Stirling numbers, 508 Hausdorff dimension, 368 inequalities for the r-Stirling numbers, high-speed computers, 73 470 higher order Bernoulli polynomials, inequalities for the Bernoulli or Euler 580 numbers, 575 higher order Euler polynomials, 558 inequality for det(S) f , 273 highly abundant, 347 infinitary amicable numbers, 69 highly powerful numbers, 334 infinitary convolution, 124 Hilbert space, 185 infinitary divisor, 48 Hilbert transform, 538 infinitary divisors, 124 Hilbert’s inequality, 354 infinitary harmonic numbers, 50 hoax number, 384 infinitary Mobius¨ function, 124 Hoheisel-Ingham theorem, 380 infinitary perfect numbers, 49 Holiday numbers, 488 infinitary sociable numbers, 75 holiday numbers, 578 infinitary totient, 287 homogeneous symmetric polynomial, infinite chess games, 390 189 infinite dimensional matrix, 127 homotopy and homology of finite infinite product, 182, 424 lattices, 148 infinite recurrences, 415 Hooley’s function, 365 infinite sums, 278, 423 Hurwitz zeta function, 423 infinitude of primes, 180 hyperelliptic equations, 507 inner product, 185 hypergeometric functions, 525 integer partitions in dominance hyperharmonic numbers, 471 order, 148 hyperperfect numbers, 50 integer-perfect numbers, 53 hyperplanes, 568 integral ideal, 293 hypothesis H,42 integral ideals, 159 Hypothesis H of Schinzel, 222 integral product, 128 Holder’s¨ theorem, 259 integral words, 151 identities, 181, 464 intermediate prime divisors, 341 incidence algebra, 145 intuitionistic fuzzy theory, 148 Incidence algebras, 139 inverse problems in Physics, 146 incidence function, 140 inversion formula, 153, 160 incidence functions, 267 inversion formulae, 100 incidence matrix, 268 inversion formulas, 462 incidence ring, 140 inversion statistic, 481 independent random variables, 511 inversion theorem, 126, 145 indicator, 179 inversion theorem of Mobius¨ type, 120 inequalities for n(x), 261 inversions, 570

628 INDEX involutions of permutations, 507 Lagrange’s theorem, 488 irrational number, 411 Lah numbers, 578 irrationality result, 254 Lah’s numbers, 478 irreducibility of polynomials, 253 Laplace integrals, 583 irreducibility of the cyclotomic Laplace method, 579 polynomial, 252 Laplace’s rule, 259 irreducible polynomial, 262, 362 large prime divisors, 341 irreducible polynomials, 192 lattice, 141 isomorphic classes of objects, 186 lattices of subgroups of a group, 139 iterated Stirling numbers, 493 law of iterated logarithms, 364 iterates, 75 lcm-closed sets S, 271 iteration, 42 least common multiple, 348 ( ) iteration of s n , 429 least nonnegative residue, 543 ϕ iteration of , 195 least prime divisor, 355 Iwasawa functions, 544 left inversion, 123 joint distribution, 403 left-factorial function, 573 Jordan block, 409 Legendre formula, 371 Jordan curve, 519 Legendre symbol, 250, 396, 547 Jordan totient, 235, 275 Legendre totient, 283, 287 Jordan totient function, 182, 214 Legendre totient of u arguments, 285 Jordan’s totient, 106, 133, 280 Legendre-Jacobi symbol, 183 Jordan’s totient function, 259 Legendre-Stirling numbers, 488 Jordan-curve, 473 Lehmer ψ-product, 128 Jordan-Nagell totient, 132, 275, 276 Lehmer’s conjecture, 212 Lehmer’s totient function, 289 Kanold’s method, 30 Lehmer, McCarthy, and Erdos¨ Kaprekar numbers, 22 theorems, 247 Kaprekar routine, 429 Leudesdorf’s theorem, 244 Kesava Menon’s totient, 281 limit distribution mod m, 389 Klee’s method, 228 limiting distribution, 540 Klee’s Mobius¨ function, 278 linear forms in logarithms, 508 Kloosterman-sums, 250 linear groups, 186 Kubilius class, 344 linear independence over Q, 263 Kummer type congruences, 552, 554, linear recurrences, 414 567 linear transformations, 185 Kurepa’s left-factorial function, 495 Liouville function, 116, 137 l-c product, 126 ”Little theorem”, Fermat, 19 l.c.m.-product, 119 local limit law, 416 Lagrange theorem, 245 locally distributive, 143

629 INDEX locally finit poset, 142 Mellin transform, 566 log-concave function, 524 Mersenne prime, 21, 22, 234 log-concavity, 577 Mersenne primes, 15, 19, 41, 42, logarithm operator, 108 222 logarithmic and identric means, 486 metabelian group, 159 logarithmic convexity, 471 methods of Voronoi, 545 logarithmic distribution function, 397 Miller-Rabin and Lucas tests, 218 logarithmic Fourier series, 398 minimal polynomial, 420 logarithmic polynomials and numbers, minimal, normal, and average order of 533 Carmichael’s function λ, 193 logarithmically concave, 470 Minkowski product, 104 lovely numbers, 69 modern computers, 24 lower closed sets, 267 modified q-Bernoulli numbers, 564 lower density, 236 modified q-Bernoulli polynomials, Lubin-Tate groups, 566 564 Lucas sequence of the second kind, modified Stirling numbers, 500 240 modular group, 272 Lucas sequence of the first kind, 240 modular idempotent numbers, 431 Lucas sequences, 36 Monica sets, 384 Lucas’ totient, 277 monoids with finite decomposition Mac Mahon’s Master theorem, 161 property, 148 magic squares, 184 multidimensional Smith’s Magma system for computational determinants, 266 algebra, 258 multiperfect numbers, 32, 35 Maillet’s theorem, 207 multiplicative inverse of t, 250 Makowski-Schinzel conjecture, 235 multiplicative semigroup, 105 Marriage theorem, 348 multiply perfect numbers, 32, 35, 71 matched asymptotic expansions, 516 Mobius¨ categories, 148 matrix-generated convolutions, 127 Mobius¨ function, 153, 154, 264, 268, maximal generalization of Fermat’s 490, 496 theorem, 190 Mobius¨ function µG , 128 maximum and minimum exponents, Mobius¨ function associated with 330 the dual poset, 157 maximum of the exponents, 355 Mobius¨ function generated by the mean value, 401 B-convolution, 116 meet semilattice, 267 Mobius¨ function of n arguments, 109 meet-closed sets, 273 Mobius¨ function of P, 140 meet-closed subset, 268 Mobius¨ function of a direct product, meet-matrix, 268 158

~~630 INDEX~~

Mobius¨ function of a generalized noncototients, 208 integer, 125 nontotient, 230 Mobius¨ function of a set of primes, nonunitary k-cycle, 76 105 nonunitary aliquot sequences, 76 Mobius¨ function of a trace monoid, nonunitary amicable pairs, 68 162 nonunitary divisors, 48 Mobius¨ function of geometric lattices, nonunitary perfect numbers, 48 148 nonunitary totient function, 287 Mobius¨ function on a subset of a normal order, 332 power set, 154 nowhere differentiable, 373, 375, 392, Mobius¨ functions of arbitrary direct 400, 417 factor sets, 135 nowhere differrentiable, 399 Mobius¨ inversion, 102–104, 148, 490 number of decimal digits, 431 Mobius¨ inversion formula, 180, 251 number of digit blocks of 11, 426 Mobius¨ inversions, 482 number of distinct prime divisors, 33, Mobius¨ system, 104 110, 180 Mobius¨ type inversion theorems, 150 number of distinct prime factors, 28, Mobius’¨ function, 184 243, 250, 265, 283, 519 Mobius’¨ function of an arithmetical number of divisors, 35, 180, 264 semigroup, 148 number of odd digits, 425 Mobius-type¨ function, 193 number of solutions of the Mobius-type¨ inversion formula, 135 congruence, 279 Mobius-type¨ inversion formulae, 125 number of square-free divisors, 130 number of terminating nines, 426 Nagell’s totient, 133, 278 number of unitary divisors, 284 Narkiewicz Mobius¨ function, 144 Narkiewiecz Mobius¨ function, 113 occurrences of the block w, 425 Narumi polynomials, 550 odd amicable pairs, 64 negative integer bases, 417 odd perfect number, 25, 28, 29, 31, 33, Newton’s formulae, 258 34, 41 Nielsen polynomials, 467 odd perfect numbers, 19, 23, 26, 31 nilpotent elements, 118, 290 open problem, 203, 209, 222, 272, Niven number, 381 349, 416 no unitary m-superperfect numbers, open problems, 31, 43, 76, 203 57 open question, 67, 209 non-central Stirling numbers, 479 operatorial theory, 469 non-commuting lifting, 162 optimization theory, 339 non-crossing set partitions, schufﬂe orbit, 490 posets, 148 orbits, 481 non-divisibility property, 202 order of a modulo n, 189

631 INDEX order of the group, 202 positive deﬁnite, 273 orthogonal projection, 394 power series, 100 orthogonality, 476, 477, 485 power-harmonic number, 44 orthonormal basis, 185 powerful, 52 oscillatory asymptotic, 524 practical numbers, 59, 359 other analogs of Lehmer’s preferential arrangements, 496 problem, 215 primality criterion, 254, 256 overlap, 390 primary pseudoperfect number, 45 prime k-tuple conjecture, 218 palindrome, 391 prime factors of n(a, b), 255 parallelotope, 420 ( ) prime factors of ϕ j (n), 198 parametric Mobius¨ inversion, 145 prime factors of the Euler Parry’s α-expansion, 415 numbers, 556 partial p, q-Stirling numbers, 486 prime ideal, 293 partial ordering, 141 prime ideal theorem, 160, 161 partial product, 128 prime number theorem, 149, 150 partially ordered sets, 105 prime-counting function π(n), partition lattice q-analogs, 481 180 partitions, 461 prime-power groups, 139 Pascal’s triangle, 379 primitive character, 553 Pell sequence, 67 perfect number, 369 primitive divisors of Lucas perfect totient number, 240 sequences, 216 periodic mapping, 554 primitive element, 420 periodic perfect numbers, 17 primitive root mod n, 251 periods of the Stirling numbers, 492 primitive root of unity, 258 Phibonacci number, 223 primitive roots, 189 Pillai arithmetic function, 182 primitive roots of unity, 138, 251 Piltz divisor function, 106, 107 primitive solutions, 197, 198 Pisot number, 415 primitive substitution, 374 Plato’s Republic, 17 probabilistic group theory, 364 pointwise product, 119 product of all e-divisors, 58 Poisson process, 341 product of all divisors, 55 poly-Bernoulli numbers, 501 product of digits, 383 polylogarithm function, 552 product of divisors, 202 polynomial ring, 552 product set, 104 polynomial sequences, 402 pseudoperfect numbers, 43 poset, 139 pseudoprime, 30 poset-theoretic generalization of Polya´ frequency sequence, 578 Smith’s theorem, 268 Polya-Vinogradov´ inequality, 554

632 INDEX quadratic character, 554 residue theorem, 478 quadratic field, 285, 293 resolution function, 367 quadratic fields, 32, 252 resultant of cyclotomic polynomials, quadratic residue, 216 262 quadri-amicable numbers, 70 reversing the digits, 429 quasi-amicable pairs, 68 Riemann hypothesis, 102, 129, 130 quasi-multiplicative functions, 270 Riemann Hypothesis, 187 quasi-multiplicativity, 116 Riemann hypothesis, 262 quasi-orthogonality, 476 Riemann zeta function, 99, 187, 263, quasi-superperfect, 41 330, 510, 527 quasiperfect numbers, 36 Riesz representation theorem, 185 queues, 464 right inversion, 123 right inversion formula of Mobius¨ Ramanujan expansions, 278 type, 123 Ramanujan identities, 536 rising factorial power, 461 Ramanujan sum, 184, 258, 277 roots of unity, 184, 393 Ramanujan sums, 125, 137 Roth’s theorem, 426 Ramanujan sums on semigroups, 277 Rudin-Schapiro sequence, 396 Ramanujan theorems, 544 Redei’s´ generalization of Euler’s Ramanujan’s sum, 265 theorem, 208 ray method, 516 reciprocal GCD and LCM matrices, saddle point method, 516, 583 272 Salem numbers, 415 recursive algorithm, 223 Schemmel totient, 184 recursive set, 32 Schemmel totient function, 229 refined Stirling numbers, 487 Schemmel’s totient, 133, 216, 276 regular S-representation, 285 Schemmel-Nagell totient, 133, 276 regular arithmetic convolution, 144 Schinzel’s H-Hypothesis, 380 regular convolution, 112 Schinzel’s conjecture, 218 regular convolutions, 183 Schinzel’s Hypothesis H, 198, 221 regular polygon, 188, 239 Schinzel-Szekeres function, 356 relatively k-prime, 136, 275 Schmidt’s subspace theorem, 418, 508 relatively prime amicable pairs, 65, 66 Schrodinger¨ operators, 391 relatively prime amicable pairs of Schur’s theorem, 260 opposite parity, 66 secant numbers, 559 remarkable identities, 282 second order Eulerian numbers, 580 representations of real numbers, 432 Selberg’s identity, 160 repunit number, 383 Selberg’s upper bound sieve, 226 residue classes, 387 self-number, 384 residue classes modulo n, 249 semi-multiplicativity, 116

633 INDEX semi-unitary divisor, 281 soluble groups, 154 semi-unitary greatest common divisor, special cyclotomic polynomials, 252 271 spectral norm of a matrix, 274 semi-unitary product, 128 square-reduced residue system, 289 semigroup approach to the Fermat and square-totient of R. Sivaramakrishnan, Euler’s theorems, 191 289 semigroups with unique factorization, squarefree integers, 33, 207, 265 150 squarefree number, 35, 46 semisimple rings, 333 squarefree numbers, 136 sequence (Bk (mod n)), 555 squarefull, 209 sequence of Eulerian numbers, 577 squarefull integers, 331 sequences of Fibonacci, Pell, and staggered sums, 429 Lucas, 59 statistic on permutations, 479 set of totients, 206 statistical convergence, 331 Shapiro-Pillai function, 200 statistical limit, 335 Shintani zeta functions, 536 Stevens totient, 280 Siegel-Walfisz theorem, 248 Stevens’ totient, 280 sieve of Eratosthenes, 338 Stieltjes transforms, 525 Sieve theory, 103 Stirling functions of the first kind, 473 sieve-theoretical problems, 355 Stirling functions of the second kind, sigmoid function, 569 473 signed Eulerian polynomial, 570 Stirling numbers associated to a signless q-Stirling numbers of the first matrix, 475 kind, 481 Stirling numbers of complex signless Bernoulli numbers, 525 argument, 487 Silverman constant, 182 Stirling numbers of complex simple groups, 157 arguments, 523 simple pole, 415 Stirling numbers of fractional order, Sita Ramaiah’s Mobius¨ function, 113 472 small prime divisors, 341 Stirling numbers of the first kind, 462, small sieve, 359 479 smallest prime totative, 248 Stirling numbers of the first kind Smarandache function, 22, 210, 237 st (n, k) associated to the Smith brothers, 384 sequence t, 474 Smith number, 383 Stirling numbers of the second kind, Smith-Jones number, 384 460, 578 sociable number of order k,75 Stirling numbers of the third kind, 464 sociable numbers, 72 Stirling polynomials, 466, 469 soluble group, 155, 157 Strassen functional, 343

634 INDEX strong log-concavity, 508 the characteristic, 511 strong regular A-convolution, 119 The exponential A-convolution, strongly q-additive, 411 118 strongly q-log-concave, 514 theorem by J. Touchard, 29 subadditive functions, 344 theorem by Touchard, 35 subblock occurrences, 422 theorem of Adams, 541 subgroup, 151, 157 theorem of Howard, 551 sublime numbers, 22 theorem of Sylvester, 25 submatrix, 269 theorem of Sylvester and subring, 142 Lipschitz, 548 subspace, 185 theoretical physics, 375 sugcd-closed, 271 theory of Mobius¨ function, 139 sum of jth powers of totatives, 242 Thue-Morse sequence, 379 sum of digits, 371, 499, 551 Thue-Morse sequence, 390 sum of digits of primes, 379 total number of prime divisors, 180 sum of digits with prime indices, 390 total number of prime factors, 26, 133, sum of divisors, 180, 264 137 sum of prime digits, 390 totatives, 179, 242 sum of totatives, 242 totient functions, 125 sum-product numbers, 383 totient-free residue classes, 206 super-lovely pair, 69 Touchard’s congruence, 495 super-multiplicative, 355 trace languages, 161 supermultiplicative, 346 trace monoids, 161 superperfect number, 58 transcendence, 391 superperfect numbers, 23, 38, 69 transcendental, 426 Suzanne sets, 384 transitive orientation, 162 Sylow p-subgroup, 155 triangular matrices, 265 Sylow subgroups, 158 triperfect numbers, 34 symmetric signed digit expansion, 422 twin dragon, 421 symmetric system, 422 twin primes, 234 symmetries and dualities, 476 twin-prime constant, 219 tangent numbers, 527, 559, 577 ud-closed, 271 Taylor expansion, 488 ultrametric Banach space, 546 te Riele’s trick, 63 unambiguous lifting, 162 Teichmuller¨ character, 554 unambiguous Mobius¨ function, Thabit’s rule, 17, 61 162 Thacker’s function, 243 unequal characters, 554 Thacker’s totient function, 289 uniform asymptotic, 524

635 INDEX uniform asymptotic expansion, 518 unitary hyperperfect numbers, 51 uniform limit distribution mod m, unitary multiply superperfect 389 numbers, 48 uniform probability measure, 343 unitary Mobius¨ function, 111, 272 uniform summability, 413 unitary perfect numbers, 45 uniformly distributed mod m, 389 unitary product, 110 uniformly distributed mod 1, 410 unitary product set, 111 uniformly distributed modulo unitary quasi superperfect numbers, 1, 397 47 uniformly periodic, 427 unitary quasi-sociable numbers, 75 unimodal, 577 unitary quasiperfect numbers, 47 unique factorization domains, 32 unitary Smith determinants, 271 unitary m-perfect numbers, 57 unitary sociable numbers, 74 unitary abundant numbers, 46 unitary superperfect numbers, 47 unitary aliquot sequences, 74 unitary totient function, 213, 266 unitary almost perfect numbers, 47 universal Bernoulli numbers, 552 unitary almost superperfect number, universal generated, 385 47 unsigned Lah numbers, 464 unitary almost-sociable numbers, 75 unsigned Stirling numbers of the ﬁrst kind, 462 unitary amicable numbers, 67 unsolved problems on generalized unitary analogue of Euler’s totient, integers, 125 242 upper density, 345 unitary analogue of Legendre’s totient, 284 valence function of ϕ, 226 unitary analogue of Nagell’s totient, vector space, 145, 286, 420, 481 281 von Mangoldt function, 160, 259 unitary analogue of Ramanujan’s von Staudt-Clausen and Voronoi type sum, 267 congruences, 554 unitary analogue of Schemmel’s von Staudt-Clausen theorem, 245, totient, 281 539 unitary analogue of Smith’s von Sterneck and the Cohen totients, determinant, 266 275 unitary analogue of the Carmichael’s conjecture, 229 Waring’s problem, 421 unitary analogues of generalized weak convergence of distribution Ramanujan sums, 277 functions, 343 unitary convolution, 112, 127 weakly-composite, 340 unitary divisor, 46 weird numbers, 43, 44 unitary harmonic numbers, 49, 50 well-distributed mod 1, 411

~~636 INDEX~~

Weyl derivative, 474 Worpitzky type formulas, 572 Weyl’s lemma, 421 wreath products, 490 Wieferich prime, 212 Wieferich primes, 540 Zeckendorf sum-of-digits function, Wiegandt convolution, 139 398 Wiener measure, 344 Zeckendorf Thue-Morse sequence, Wilson quotient, 540, 547 399 Wilson theorem, 574 zero divisors, 118 Wilson’s theorem, 191, 488, 542 Zsigmondy-Birkhoff-Vandiver Worpitzky identity, 568 theorem, 202

~~637~~