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Handbook of Number Theory Ii HANDBOOK OF NUMBER THEORY II by 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. ISBN 1-4020-2546-7 (HB) ISBN 1-4020-2547-5 (e-book) Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper 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 . .............
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