Handbook of Number Theory I by József Sándor Babes-Bolyai University of Cluj, Cluj-Napoca, Romania Dragoslav S. Mitrinović formerly of the University of Belgrade, Servia and Borislav Crstici formerly of the Technical University of Timisoara, Romania A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-10 1-4020-4215-9 (HB) ISBN-13 978-1-4020-4215-7 (HB) ISBN-10 1-4020-3658-2 (e-book) ISBN-13 978-1-4020-3658-3 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springeronline.com Printed on acid-free paper 1st ed. 1995. 2nd printing All Rights Reserved © 2006 Springer 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. TABLE OF CONTENTS PREFACE ....................................................................................... xxv BASIC SYMBOLS ............................................................................ 1 BASIC NOTATIONS.......................................................................... 2 Chapter I EULER’S ϕ-FUNCTION..................................................................... 9 § I. 1 Elementary inequalities for ␸ .............................................. 9 § I. 2 Inequalities for ␸(mn)....................................................... 9 § I. 3 Relations connecting ␸, ␴, d ............................................... 10 § I. 4 Inequalities for Jk , ␴k , ␺ k ................................................... 11 § I. 5 Unitary analogues of Jk , ␴k , d ............................................. 12 § I. 6 Composition of ␸, ␴, ␺ ..................................................... 13 § I. 7 Composition of ␴, ␸ ......................................................... 13 § I. 8 On the function n/␸(n) ...................................................... 14 § I. 9 Minimum of ␸(n)/n for consecutive values of n ....................... 15 § I.10 On ␸(n + 1)/␸(n)........................................................... 16 § I.11 On (␸(n + 1), ␸(n)) ......................................................... 18 § I.12 On (n, ␸(n)) ................................................................... 18 § I.13 The difference of consecutive totients ................................... 19 § I.14 Nonmonotonicity of ␸ .(Ameasure) .................................... 19 § I.15 Nonmonotonicity of Jk ..................................................... 20 § I.16 Number of solutions of ␸(x) = n! ........................................ 20 § I.17 Number of solutions of ␸(x) = m ........................................ 21 § I.18 Number of values of ␸ less than or equal to x ......................... 22 § I.19 On composite n with ␸(n)|(n − 1) (Lehmer’s conjecture) ........... 23 § I.20 Number of composite n ≤ x with ␸(n)|(n − 1) ........................ 24 § I.21 ␸(n) ........................................................................ 24 n≤x k § I.22 f · ␸(k) ............................................................ 25 k≤n n 3 § I.23 On ␸(n) − x2 ........................................................ 25 ␲2 n≤x § I.24 On ␸(n)/n ................................................................ 27 n≤x vi Table of Contents § k+1 I.25 On Jk (n) − x /(k + 1)␨(k + 1) ................................... 28 n≤x § I.26 An expansion of Jk .......................................................... 29 § /␸ I.27 On n≤x 1 (n) and related questions .................................. 29 § ␸ − I.28 p≤x (p 1) for p prime ............................................... 30 § ␸ , I.29 On n≤x ( f(n)) f a polynomial ....................................... 31 § ␸∗ , ␸ ␸ + I.30 n≤x (n) n≤x (n) (n k) and related results ................. 31 § I.31 Asymptotic formulae for generalized Euler functions ................ 32 § ␸ , = I.32 On (x n) m≤x,(m,n)=1 1 and on Jacobstahl’s arithmetic function ........................................................................ 33 § I.33 On the iteration of ␸ ........................................................ 34 § I.34 Iterates of ␸ and the order of ␸(k)(n)/␸(k+1)(n)........................ 35 § I.35 Normal order of ␻(␸(n)) ................................................... 36 Chapter II THE ARITHMETICAL FUNCTION d(n), ITS GENERALIZATIONS AND ITS ANALOGUES.................................................................................. 39 § II. 1 The divisor functions at consecutive integers .......................... 39 § II. 2 On d(n + i1) > ···> d(n + ir ) .......................................... 40 § II. 3 Relations connecting d, ␻, , dk ......................................... 41 § II. 4 On d(mn) ..................................................................... 42 § II. 5 An inequality for dk (n) ..................................................... 42 § II. 6 Majorization for log d(n)/ log 2 .......................................... 42 § II. 7 max d(n) and max(d(n), d(n + 1)) and generalizations .............. 44 n≤x n≤x § II. 8 Highly composite, superior highly composite, and largely composite numbers .......................................................... 45 § II. 9 Congruence property of d(n) .............................................. 47 § II.10 (x) = d(n) − x log x − (2␥ − 1)x ............................... 47 n≤x § II.11 d(p − 1), p prime ..................................................... 49 p≤x § II.12 k (x) = dk (n) − x · Pk−1(log x), k ≥ 2 ........................... 51 n≤x § 2 II.13 dk (n) ..................................................................... 55 n≤x § II.14 On (g ∗ dk )(n) .......................................................... 55 n≤x § II.15 3(x) .......................................................................... 56 § II.16 The divisor problem in arithmetic progressions ...................... 57 § II.17 On 1/dk (n) .............................................................. 59 n≤x Table of Contents vii § II.18 Average order of dk (n) over integers free of large prime factors ......................................................................... 60 § II.19 On a sum on dk and Legendre’s symbol ................................ 60 § II.20 A sum on dk , d and ␻ ...................................................... 61 § II.21 On d(n) · d(n + N) and related problems ......................... 61 n≤x § II.22 On dk (n) · d(n + 1) and related questions ......................... 63 n≤x § II.23 Iteration of d ................................................................. 65 § II.24 On d( f (n)) and d(d( f (n))), f a polynomial .......................... 66 § II.25 On d(n2 + a) and d(m2 + n2) ................................ 67 n≤x m,n≤x § II.26 d(| f (r, s)|), f (x, y)abinary cubic form .................... 68 | f (r,s)|≤N § II.27 Weighted divisor problem ................................................. 68 § II.28 On d(n − kt ) .......................................................... 69 k<n1/t § II.29 Divisor sums on squarefree or squarefull integers .................... 69 § II.30 Exponential divisors ....................................................... 71 § II.31 Bi-unitary divisors .......................................................... 72 § II.32 Sums over d(n) · ␻(n), d(n)/␻(n), ␻(d(n)), (d(n)) ................ 72 § II.33 d(a(n)), a(n) the number of abelian groups with n n≤x elements ...................................................................... 73 § II.34 d(n)inshort intervals ...................................................... 73 § II.35 Number of distinct values of d(n) for 1 ≤ n ≤ x ..................... 74 § II.36 On the distribution function of d(n) ..................................... 74 § II.37 On (nd(n), ␴(n)) = 1 ...................................................... 75 § II.38 Average value for the number of divisors of sums a + b ............ 75 Chapter III SUM-OF-DIVISORS FUNCTION, GENERALIZATIONS, ANALOGUES; PERFECT NUMBERS AND RELATED PROBLEMS ................................ 77 § III. 1 Elementary inequalities on ␴(n) and ␴(n)/n ......................... 77 § III. 2 On ␴(n)/n log log n ........................................................ 79 § ␴ / k III. 3 On k (n) n................................................................. 80 § III. 4 ␴(n), ␴(n), ␴(n) ................................... 81 n≤x n≤x,p|n n≤x,(n,k)=1 ␴(n) § III. 5 Sums over ............................................................ 82 n § III. 6 Sums over ␴k (n) ............................................................ 83 § III. 7 On sums over ␴−␣( f (n)), f a polynomial (0 < ␣ < 1) ............. 84 viii Table of Contents § III. 8 On ␴( f (n)), f a polynomial ......................................... 85 n≤x § III. 9 Sums on ␴␣(n), ␴␤(n + k)................................................. 85 § III.10 Inequalities connecting ␴k , d, ␥, .................................... 86 § III.11 Sums over ␴(p − 1), p a prime ......................................... 87 § III.12 On ␴(mn) ................................................................... 87 § III.13 On ␴(n) ≥ 4␸(n) .......................................................... 88 § III.14 On ␴(n + i)/␴(n + i − 1) and related theorems .................... 88 § III.15 On ␴(␴(n)); ␴∗(␴∗(n))