HANDBOOK OF NUMBER THEORY II

by

J. Sändor Babes-Bolyai University ofCluj

Department of Mathematics and Computer Science Cluj-Napoca, Romania

and

B. Crstici formerly the Technical University of Timisoara

Timisoara Romania

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

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 y-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 MÖBIUS FUNCTION 99 2.1 Introduction 99

1

CONTENTS

2.2 Möbius functions generated by arithmetical products (or convolutions) 106 1 Möbius functions defined by Dirichlet products 106 2 Unitary Möbius functions 110 3 Bi-unitary Möbius function 111 4 Möbius functions generated by regulär convolutions . . . . 112 5 K-convolutions and Möbius functions. B convolution . . . . 114 6 Exponential Möbius functions 117 7 l.c.m.-product (von Sterneck-Lehmer) 119 8 Golomb-Guerin convolution and Möbius function 121 9 max-product (Lehmer-Buschman) 122 10 Infinitary convolution and Möbius function 124 11 Möbius function of generalized (Beurling) integers 124 12 Lucas-Carlitz (1-c) product and Möbius functions 125 13 Matrix-generated convolution 127

2.3 Möbius function generalizations by other number theoretical considerations 129 1 Apostol's Möbius function of order k 129 2 Sastry's Möbius function 130 3 Möbius functions of Hanumanthachari and

Subrahmanyasastri 132 4 Cohen's Möbius functions and totients 134 5 Klee's Möbius function and totient 135 6 Möbius functions of Subbarao and Harris; Tanaka;

and Venkataraman and Sivaramakrishnan 136 7 Möbius functions as coefficients of the cyclotomic

polynomial 138 2.4 Möbius functions of posets and lattices 139

1 Introduction, basic results 139 2 Factorable incidence functions, applications 143 3 Inversion fheorems and applications 145 4 Möbius functions on Eulerian posets 146 5 Miscellaneous results 148

2.5 Möbius functions of arithmetical semigroups, free groups, finite groups, algebraic number fields, and trace monoids 148 1 Möbius functions of arithmetical semigroups 148 2 Fee abelian groups and Möbius functions 151 3 Möbius functions of finite groups 154

2

CONTENTS

4 Möbius functions of algebraic number and function-fields 159

5 Trace monoids and Möbius 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 <p 180 3 Infinite series and products involving

<p, Pillai's (Cesäro's) arithmetic functions 181 4 Enumeration problems on congruences, directed

graphs, magic Squares 183 5 Fourier coefficients of even functions (mod n) 184 6 Algebraic independence of arithmetic functions 185 7 Algebraic and analytic application of totients 186 8 ^-convergence of Schoenberg 187

3.2 Congruence properties of Euler's totient and related functions . . . . 188 1 Euler's divisibility theorem 188 2 Carmichael'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 5 Divisibility properties of iteration of <p 195 6 Congruence properties of <p and related functions 201 7 Euler's totient in residue classes 204 8 Prime totatives 206 9 The dual of <p, 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 <p(x + k) = <p(x) 216 2 <p(x + k) = 2<p(x + k) = <p(x) + <p(k) and related

equations 221 3 Equation <p(x) = k, Carmichael's conjecture 225 4 Equations involving cp and other arithmetic functions . . . . 230 5 The composition of (p 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 2 Divisibility properties 253 3 The coefhcients of cyclotomic polynomials 256 4 Miscellaneous results 261

3.6 Matrices and determinants connected with q> 263 1 Smith's determinant 263 2 Poset-theoretic generalizations 266 3 Factor-closed, gcd-closed, lcm-closed sets, and

related determinants 270 4 Inequalities 273

3.7 Generalizations and extensions of Euler's totient 275 1 Jordan, Jordan-Nagell, von Sterneck, Cohen-totients . . . . 275 2 Schemmel, Schemmel-Nagell, Lucas-totients 276 3 Ramanujan'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 8 Legendre'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 ofa 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 A function 360 11 Extensions of the Erdös 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 4 Niven numbers 381 5 Smith numbers 383 6 Seif numbers 384 7 The sum-of-digits function in residue classes 387 8 Thue-Morse and Rudin-Shapiro sequences 390 9 g-additive and g-multiplicative functions 401 10 Uniform - and well - distributions ofceSq(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 Symmetrie 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 BernouUi and Euler numbers 525 1 Definitions, basic properties of BernouUi numbers

and polynomials 525 2 Identities 534 3 Congruences for BernouUi numbers and polynomials.

Eulerian numbers and polynomials 539 4 Estimates and inequalities 574

References 585

Index 619

6