Handbook of number theory II 2006

HANDBOOK OF NUMBER THEORY II by J S´andor 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 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 C Printed in the Netherlands Contents PREFACE BASIC SYMBOLS BASIC NOTATIONS 10 PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES 1.1 Introduction 1.2 Some historical facts 1.3 Even perfect numbers 1.4 Odd perfect numbers 1.5 Perfect, multiperfect and multiply perfect numbers 1.6 Quasiperfect, almost perfect, and pseudoperfect numbers 1.7 Superperfect and related numbers 1.8 Pseudoperfect, weird and harmonic numbers 1.9 Unitary, bi-unitary, infinitary-perfect and related numbers 1.10 Hyperperfect, exponentially perfect, integer-perfect and γ -perfect numbers 1.11 Multiplicatively perfect numbers 1.12 Practical numbers 1.13 Amicable numbers 1.14 Sociable numbers 15 15 16 20 23 32 36 38 42 45 50 55 58 60 72 References 77 GENERALIZATIONS AND EXTENSIONS OF THE ă MOBIUS FUNCTION 2.1 Introduction 99 99 CONTENTS 2.2 2.3 2.4 2.5 Măobius functions generated by arithmetical products (or convolutions) Măobius functions defined by Dirichlet products Unitary Măobius functions Bi-unitary Măobius function Măobius functions generated by regular convolutions K -convolutions and Măobius functions B convolution Exponential Măobius functions l.c.m.-product (von Sterneck-Lehmer) Golomb-Guerin convolution and Măobius function max-product (Lehmer-Buschman) 10 Infinitary convolution and Măobius function 11 Măobius function of generalized (Beurling) integers 12 Lucas-Carlitz (l-c) product and Măobius functions 13 Matrix-generated convolution Măobius function generalizations by other number theoretical considerations Apostols Măobius function of order k Sastrys Măobius function Măobius functions of Hanumanthachari and Subrahmanyasastri Cohens Măobius functions and totients Klees Măobius function and totient Măobius functions of Subbarao and Harris; Tanaka; and Venkataraman and Sivaramakrishnan Măobius functions as coefficients of the cyclotomic polynomial Măobius functions of posets and lattices Introduction, basic results Factorable incidence functions, applications Inversion theorems and applications Măobius functions on Eulerian posets Miscellaneous results Măobius functions of arithmetical semigroups, free groups, finite groups, algebraic number fields, and trace monoids Măobius functions of arithmetical semigroups Fee abelian groups and Măobius functions Măobius functions of finite groups 106 106 110 111 112 114 117 119 121 122 124 124 125 127 129 129 130 132 134 135 136 138 139 139 143 145 146 148 148 148 151 154 CONTENTS Măobius functions of algebraic number and function-fields Trace monoids and Măobius functions References 159 161 163 THE MANY FACETS OF EULER’S TOTIENT 3.1 Introduction The infinitude of primes Exact formulae for primes in terms of ϕ Infinite series and products involving ϕ, Pillai’s (Ces`aro’s) arithmetic functions Enumeration problems on congruences, directed graphs, magic squares Fourier coefficients of even functions (mod n) Algebraic independence of arithmetic functions Algebraic and analytic application of totients ϕ-convergence of Schoenberg 3.2 Congruence properties of Euler’s totient and related functions Euler’s divisibility theorem Carmichael’s function, maximal generalization of Fermat’s theorem Gauss’ divisibility theorem Minimal, normal, and average order of Carmichael’s function Divisibility properties of iteration of ϕ Congruence properties of ϕ and related functions Euler’s totient in residue classes Prime totatives The dual of ϕ, noncototients 10 Euler minimum function 11 Lehmer’s conjecture, generalizations and extensions 3.3 Equations involving Euler’s and related totients Equations of type ϕ(x + k) = ϕ(x) ϕ(x + k) = 2ϕ(x + k) = ϕ(x) + ϕ(k) and related equations Equation ϕ(x) = k, Carmichael’s conjecture Equations involving ϕ and other arithmetic functions The composition of ϕ and other arithmetic functions Perfect totient numbers and related results 179 179 180 180 181 183 184 185 186 187 188 188 189 191 193 195 201 204 206 208 210 212 216 216 221 225 230 234 240 CONTENTS 3.4 3.5 3.6 3.7 The totatives (or totitives) of a number Historical notes, congruences The distribution of totatives Adding totatives Adding units (mod n) Distribution of inverses (mod n) Cyclotomic polynomials Introduction, irreducibility results Divisibility properties The coefficients of cyclotomic polynomials Miscellaneous results Matrices and determinants connected with ϕ Smith’s determinant Poset-theoretic generalizations Factor-closed, gcd-closed, lcm-closed sets, and related determinants Inequalities Generalizations and extensions of Euler’s totient Jordan, Jordan-Nagell, von Sterneck, Cohen-totients Schemmel, Schemmel-Nagell, Lucas-totients Ramanujan’s sum Klee’s totient Nagell’s, Adler’s, Stevens’, Kesava Menon’s totients Unitary, semi-unitary, bi-unitary totients Alladi’s totient Legendre’s totient Euler totients of meet semilattices and finite fields 10 Nonunitary, infinitary, exponential-totients 11 Thacker’s, Leudesdorf’s, Lehmer’s, Golubev’s totients Square totient, core-reduced totient, M-void totient, additive totient 12 Euler totients of arithmetical semigroups, finite groups, algebraic number fields, semigroups, finite commutative rings, finite Dedekind domains 242 242 246 248 249 250 251 251 253 256 261 263 263 266 270 273 275 275 276 277 278 278 281 282 283 285 287 289 292 References 295 SPECIAL ARITHMETIC FUNCTIONS CONNECTED WITH THE DIVISORS, OR WITH THE DIGITS OF A NUMBER 4.1 Introduction 329 329 CONTENTS 4.2 4.3 Special arithmetic functions connected with the divisors of a number Maximum and minimum exponents The product of exponents Arithmetic functions connected with the prime power factors Other functions; the derived sequence of a number The consecutive prime divisors of a number The consecutive divisors of an integer Functional limit theorems for the consecutive divisors Miscellaneous arithmetic functions connected with divisors Arithmetic functions of consecutive divisors 10 Hooley’s function 11 Extensions of the Erdăos conjecture (theorem) 12 The divisors in residue classes and in intervals 13 Divisor density and distribution (mod 1) on divisors 14 The fractal structure of divisors 15 The divisor graphs Arithmetic functions associated to the digits of a number The average order of the sum-of-digits function Bounds on the sum-of-digits function The sum of digits of primes Niven numbers Smith numbers Self numbers The sum-of-digits function in residue classes Thue-Morse and Rudin-Shapiro sequences q-additive and q-multiplicative functions 10 Uniform - and well - distributions of αsq (n) 11 The G-ary digital expansion of a number 12 The sum-of-digits function for negative integer bases 13 The sum-of-digits function in algebraic number fields 14 The symmetric signed digital expansion 15 Infinite sums and products involving the sum-of-digits function 16 Miscellaneous results on digital expansions References 330 330 332 334 336 337 342 343 345 349 360 363 363 366 367 369 371 371 376 379 381 383 384 387 390 401 410 414 417 418 421 423 427 433 CONTENTS STIRLING, BELL, BERNOULLI, EULER AND EULERIAN NUMBERS 5.1 Introduction 5.2 Stirling and Bell numbers Stirling numbers of both kinds, Lah numbers Identities for Stirling numbers Generalized Stirling numbers Congruences for Stirling and Bell numbers Diophantine results Inequalities and estimates 5.3 Bernoulli and Euler numbers Definitions, basic properties of Bernoulli numbers and polynomials Identities Congruences for Bernoulli numbers and polynomials Eulerian numbers and polynomials Estimates and inequalities 459 459 459 459 464 469 488 507 508 525 525 534 539 574 References 585 Index 619 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 virtually almost all major topics as Discrete mathematics, Combinatorial theory, Numerical analysis, Finite difference calculus, Probability theory; and such classical fields of mathematics as Algebra, Geometry, and Mathematical analysis Some aspects of Chapter and 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 Mitrinovi´c of Belgrade 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 Măobius 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 Combinatorics, 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 PREFACE We have used a wealth of literature, consisting of books, monographs, journals, separates, reviews from Mathematical Reviews and from Zentralblatt făur Mathematik, 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 communications 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 Bret`eche, B C Carlson, C Cooper, G L Cohen, M Deaconescu, R Dussaud, M Drmota, J Desarmenien, K Dilcher, P Erdăos, P D T A Elliott, M Eie, S Finch, K Ford, J B Friedlander, J Feh´er, A A Gioia, A Grytczuk, K Gyăory, 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 Ivi´c, H Iwata, K.-H Indlekofer, F Halter-Koch, H.-J Kanold, M Kishore, I Katai, P A Kemp, E Krăatzel, 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 Manstaviˇcius, N S Mendelsohn, J.-L ˇ Porubsk´y, L Nicolas, E Neuman, W G Novak, H Niederhausen, C Pomerance, S Panaitopol, J E Peˇcari´c, Zs P´ales, 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 Sarkăozy, at, J O Shallit, K B Stolarsky, A Schinzel, R Sivaramakrishnan, J Sur´anyi, T Sal´ B E Sagan, I Sh Slavutskii, F Schipp, V E S Szab´o, L T´oth, 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 W´ojtowicz 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 R´enyi 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 INDEX composition of some arithmetic functions, 39 computational evidence, 42 computer algebra, 486 conductor, 553 congruence of Kummer, 541 congruence of Voronoi, 541 congruence properties, 477 congruence properties of σ (m) (n), 42 congruence properties of Euler’s totient, 201 congruence property, 188 congruences for Genocchi numbers, 541 congruences for higher order Bernoulli numbers, 549 congruences for Stirling numbers, 488 conjecture, 42, 47, 51–53, 59, 195, 229, 235, 241, 348, 382, 400, 494, 499, 556, 558, 573 conjecture of Sierpinski, 229 conjectured, 46, 352 conjectures, 199, 216, 237, 557 conjugacy classes, 157 conjugate Bernoulli numbers, 538 conjugate Bernoulli polynomials, 538 consecutive divisors, 342 consecutive prime factors, 337 constants, 181 continued fractions, 432, 537, 561 continued powers and roots, 432 convex polytope, 559 convolution of incidence functions, 139 convolution on the set of sequences, 537 convolutions with unbounded unity, 128 core, 208 Cauchy product, 117, 122, 128 Cauchy’s integral formula, 579 central factorial numbers, 479 central limit theorem, 344, 402, 423 chain, 72, 369 chains, 141, 144 change of the sum of digits by multiplication, 378 character group, 151 characteristic equation, 240 characteristic polynomial, 414 characters of finite abelian groups, 151 Chebysheff’s theorem, 207 circle of convergence, 415 circulant matrix, 183 circular permutations, 192 classical Măobius function, 99, 128, 135 code, 179, 396 coefficients of cyclotomic polynomials, 256 Cohen totient ϕ S , 290 combinatorial and statistical applications, 480 combinatorial applications, 107, 477 combinatorics on words, 390 common generalization of Liouville’s function and the Măobius function, 137 commutative group, 106 commutative ring, 110, 121 commutative semigroup, 117, 118 commutator subgroup, 158 complete asymptotic expansion, 520 complete symmetric functions, 471 complex bases, 417 complex variables, 381 composition of ϕ and other functions, 234 623 INDEX derangements and cycle indicators, 507 derived k-cycle, 337 derived sequence, 337 descents, 570 determinant, 269 determinants, 265 determinants involving q-Stirling numbers, 483 Dibag’s von Staudt-Clausen theorem, 553 Dickman function, 359 Dickson’s conjecture, 221 Dickson’s theorem, 30 Dickson-Eulerian numbers, 571 Dickson-Stirling numbers, 477 difference operator, 123 differential equation, 567 differential equations, 469 differential geometry, 390 digits of Fibonacci numbers, 430 Diophantine approximations, 263 Diophantine equations, 28, 61 Diophantine Equations, 107 diophantine equations, 219 direct factor set, 135 directed graph, 183 Dirichlet convolution, 112, 117, 127 Dirichlet density, 152 Dirichlet inverse, 120 Dirichlet inversion, 123 Dirichlet product, 106, 109, 119, 132 Dirichlet series, 100, 131, 243, 423 Dirichlet set, 206 Dirichlet’s theorem on arithmetical progressions, 210 discrete analog of integration by parts, 145 discriminator, 211 core-reduced residue system, 290 core-reduced totient of J S´andor and R Sivaramakrishnan, 290 cosets, 151 counterexample to Carmichael’s conjecture, 228 Criptography, 15 cross-convolution, 113 cumulative distribution function, 397 cycle, 72, 427 cycles, 73, 75, 183, 462, 479, 491 cyclic homology, 570 cyclotomic Bernoulli numbers, 539 cyclotomic field, 539 cyclotomic fields, 556 cyclotomic polynomial, 27, 138, 139, 203, 251, 256, 506, 571 cyclotomic polynomials, 24, 261 Daykin’s Măobius inversion formula, 110 Dedekind arithmetic function, 39 Dedekind arithmetical function, 231, 284 Dedekind domains, 191 Dedekind zeta-function, 293 deficient numbers, 17, 43 degenerate Eulerian numbers, 572 degenerate weighted Stirling numbers, 476 degree of resolution, 367 dense, 331 density, 221 density of e-perfect numbers, 52 density of k-perfect numbers, 33 density of amicable pairs, 65 density of friendly pairs, 70 density of harmonic numbers, 44 density of multiamicable numbers, 71 density of odd perfect numbers, 30 624 INDEX Euclidean norm of a matrix, 274 Euler factor, 23, 30 Euler functions of finite groups, 293 Euler gamma function, 563 Euler l.c.m sequence, 211 Euler maximal function, 210 Euler minimum function, 210 Euler numbers, 527 Euler quotients, 538 Euler totient, 40, 496 Euler totient of matrices, 278 Euler’s divisibility theorem, 188, 538 Euler’s function, 22 Euler’s gamma function, 523 Euler’s methods, 61 Euler’s theorem, 24 Euler’s totient, 67, 71, 106, 134, 137, 179, 184, 497, 544, 555 Euler’s totient function, 269, 272 Euler-MacLaurin summation formula, 547 Euler-Maclaurin summation formula, 576 Eulerian functions, 571 Eulerian numbers, 567 Eulerian numbers of higher order, 571 Eulerian polynomials, 477, 568 Eulerian posets, 146 even functions (mod n), 184 even perfect number, 19, 21, 22, 32, 54 even perfect numbers, 20, 22, 23, 44, 59 even-odd amicable pair, 65 exponential analogue à(e) of the Măobius function, 118 exponential Bell polynomials, 477 exponential convolution, 117 exponential cycles, 75 exponential divisor, 51 distance geometry, 486 distinct divisors, 347 distribution (mod 1) on divisors, 366 distribution function, 242, 345, 351, 353, 402, 524 distribution functions, 355 distribution in residue classes, 206 distribution of Bernoulli numbers modulo primes, 556 distribution of primes, 211 distribution of totatives, 246 divided differences, 469 divisibility algorithm, 191 divisor density, 366 divisorial graph, 369 divisors in residue classes, 363 divisors of factorials, 354 Dobinski’s formula, 461 double series, 101 double-factorial, 483 dynamical system, 367 Egyptean fractions, 16 eigenspace, 394 eigenvalue, 393, 409 eigenvectors, 393 eingenvalues, 393 Eisenstein series, 544 elementary Abelian, 156 elementary abelian, 159 elementary symmetric function, 510 epimorphisms, 158 equidistributed (mod 1), 366 Erdăos-Kac theorem, 388 Erdăos-Turan-Koksma inequality, 405 Erdăos-Wintner theorem, 401 ergodic theory, 412 Euclid’s ”Elements”, 16 Euclid-Euler theorem, 21 Euclidean division, 570 625 INDEX exponential inverse, 118 exponential operator, 108 exponential sum of digit counting function, 379 exponential sums, 261 exponential totient function, 119, 288 exponentially A-multiplicative functions, 119 exponentially multiplicative functions, 118 extended Nagell totient, 132 extension of Nagell’s (and Alder’s) totient, 279 Fibonacci sequence, 22 Fibonacci, or Lucas numbers, 204 finite p-group, 293 finite alphabet, 161, 374 finite automata, 390 finite field, 192, 286 finite group, 154, 490 finite permutation groups, 193 finite poset, 154 finite recurrences, 414 flag, 481 formal languages, 161 formal Laurent series, 488 formal power series, 162, 426 Fourier analysis, 415 Fourier coefficients, 184, 544 Fourier expansions, 531 Fourier transforms, 103 fractal dimension, 367 fractional parts of the Bernoulli numbers, 540 Frattini chief factor, 155 free abelian group, 151 frequency of a block, 421 friendly pair, 70 Frobenius congruences, 544 Frobenius’ classical theorem, 156 function λ, 189 function by Jacobsthal, 247 functional limit theorems, 416 fundamental domains, 421 factor-closed set, 267 factorable function, 143 factorable functions, 127 factorial base, 386 factorial numbers, 478 factorial ring, 106 factorization methods of integers, 256 factorization of Bernoulli numerators, 556 falling factorial, 478 falling factorial power, 460 Fermat number, 223 Fermat numbers, 254 Fermat prime, 27, 191, 239, 527 Fermat primes, 225, 236 Fermat quotient, 548 Fermat quotients, 530 Fermat’s ”little theorem”, 15 Fermat’s divisibility theorem, 179 Fermat’s little theorem, 488 Fermat’s quotient, 211 fermionic oscillators, 487 Fibonacci and Lucas numbers, 182, 498 Fibonacci numbers, 398 Fibonacci or Lucas sequences, 239 Galois field, 481 Gauss norm, 547 Gauss sum, 566 Gauss’ theorem, 193 Gaussian integer, 263 Gaussian integers, 32, 404 Gaussian limit law, 399 Gaussian prime, 26, 263 626 INDEX generalized Thabit’s formulae, 61 generalized totient function, 269 generalized totient functions, 137 generating function, 460, 461, 480, 483, 505, 578 generating functions, 474, 476, 484, 570 Genocchi numbers, 532 Genocchi numbers of order k, 533 geometrical investigations of congruence theorems, 189 Gill’s complexity class, 36 golden number, 486 Golomb-Guerin totient function, 288 Golubev totient functions, 289 good lattice points modulo composite numbers, 345 graceful graphs, 369 graded poset, 146 Gram’s determinant, 420 gravaritm, 201 greatest common divisor, 263 greatest prime divisor of m, 255 greatest prime factor, 356 group of units of the ring Zn , 249 groups acting on functions, 490 groups acting on graphs, 490 Guerin analogue of the Măobius function, 122 GCD matrices associated with arithmetical functions, 273 gcd-closed matrices, 270 gcd-closed set, 270 gcud-closed, 271 Gegenbauer’s theorem, 192 general theorem of log-convexity, 511 generalization of Euler’s theorem, 191, 192 generalization of Euler’s totient, 246 generalization of finite differences, 475 generalization of Măobius function, 115 generalization of the Măobius function, 137 generalizations of Eulers totient, 291 generalizations of Klee’s totient, 278 generalizations of Ramanujan’s sum, 277 generalizations of Ramanujans sum to matrices, 278 generalizations of the Măobius function, 139 generalized q-Bernoulli numbers, 562 generalized q-Euler numbers, 562 generalized arithmetical functions, 140 generalized Bernoulli numbers, 553 generalized Eulerian numbers, 571 generalized harmonic numbers, 488 generalized Liouville function, 114 generalized Măobius function, 107, 108, 135, 138 generalized Măobius functions, 134 generalized Măobius inversion, 104 generalized primes, 150 generalized Ramanujan sum, 114 generalized Stirling numbers, 469 H-Hypothesis by Schinzel, 229 Hall π -subgroup, 158 Hankel contour, 523 Hankel integral, 581 Hankel matrix, 483 Hardy-Littlewood circle method, 261 Hardy-Littlewood conjecture, 219 Harmonic analysis, 415 harmonic number, 466 harmonic numbers, 43, 44, 472 627 INDEX inequalities for Stirling numbers, 508 inequalities for the r -Stirling numbers, 470 inequalities for the Bernoulli or Euler numbers, 575 inequality for det(S) f , 273 infinitary amicable numbers, 69 infinitary convolution, 124 infinitary divisor, 48 infinitary divisors, 124 infinitary harmonic numbers, 50 infinitary Măobius function, 124 infinitary perfect numbers, 49 infinitary sociable numbers, 75 infinitary totient, 287 infinite chess games, 390 infinite dimensional matrix, 127 infinite product, 182, 424 infinite recurrences, 415 infinite sums, 278, 423 infinitude of primes, 180 inner product, 185 integer partitions in dominance order, 148 integer-perfect numbers, 53 integral ideal, 293 integral ideals, 159 integral product, 128 integral words, 151 intermediate prime divisors, 341 intuitionistic fuzzy theory, 148 inverse problems in Physics, 146 inversion formula, 153, 160 inversion formulae, 100 inversion formulas, 462 inversion statistic, 481 inversion theorem, 126, 145 inversion theorem of Măobius type, 120 inversions, 570 Haukkanen Măobius function, 114 Hausdorff dimension, 368 high-speed computers, 73 higher order Bernoulli polynomials, 580 higher order Euler polynomials, 558 highly abundant, 347 highly powerful numbers, 334 Hilbert space, 185 Hilbert transform, 538 Hilbert’s inequality, 354 hoax number, 384 Hoheisel-Ingham theorem, 380 Holiday numbers, 488 holiday numbers, 578 homogeneous symmetric polynomial, 189 homotopy and homology of finite lattices, 148 Hooley’s function, 365 Hurwitz zeta function, 423 hyperelliptic equations, 507 hypergeometric functions, 525 hyperharmonic numbers, 471 hyperperfect numbers, 50 hyperplanes, 568 hypothesis H , 42 Hypothesis H of Schinzel, 222 Hăolders theorem, 259 identities, 181, 464 incidence algebra, 145 Incidence algebras, 139 incidence function, 140 incidence functions, 267 incidence matrix, 268 incidence ring, 140 independent random variables, 511 indicator, 179 inequalities for n (x), 261 628 INDEX Lagrange’s theorem, 488 Lah numbers, 578 Lah’s numbers, 478 Laplace integrals, 583 Laplace method, 579 Laplace’s rule, 259 large prime divisors, 341 lattice, 141 lattices of subgroups of a group, 139 law of iterated logarithms, 364 lcm-closed sets S, 271 least common multiple, 348 least nonnegative residue, 543 least prime divisor, 355 left inversion, 123 left-factorial function, 573 Legendre formula, 371 Legendre symbol, 250, 396, 547 Legendre totient, 283, 287 Legendre totient of u arguments, 285 Legendre-Jacobi symbol, 183 Legendre-Stirling numbers, 488 Lehmer ψ-product, 128 Lehmer’s conjecture, 212 Lehmers totient function, 289 Lehmer, McCarthy, and Erdăos theorems, 247 Leudesdorf’s theorem, 244 limit distribution mod m, 389 limiting distribution, 540 linear forms in logarithms, 508 linear groups, 186 linear independence over Q, 263 linear recurrences, 414 linear transformations, 185 Liouville function, 116, 137 ”Little theorem”, Fermat, 19 local limit law, 416 locally distributive, 143 involutions of permutations, 507 irrational number, 411 irrationality result, 254 irreducibility of polynomials, 253 irreducibility of the cyclotomic polynomial, 252 irreducible polynomial, 262, 362 irreducible polynomials, 192 isomorphic classes of objects, 186 iterated Stirling numbers, 493 iterates, 75 iteration, 42 iteration of s(n), 429 iteration of ϕ, 195 Iwasawa functions, 544 joint distribution, 403 Jordan block, 409 Jordan curve, 519 Jordan totient, 235, 275 Jordan totient function, 182, 214 Jordan’s totient, 106, 133, 280 Jordan’s totient function, 259 Jordan-curve, 473 Jordan-Nagell totient, 132, 275, 276 Kanold’s method, 30 Kaprekar numbers, 22 Kaprekar routine, 429 Kesava Menons totient, 281 Klees method, 228 Klees Măobius function, 278 Kloosterman-sums, 250 Kubilius class, 344 Kummer type congruences, 552, 554, 567 Kurepa’s left-factorial function, 495 l-c product, 126 l.c.m.-product, 119 Lagrange theorem, 245 629 INDEX Mellin transform, 566 Mersenne prime, 21, 22, 234 Mersenne primes, 15, 19, 41, 42, 222 metabelian group, 159 methods of Voronoi, 545 Miller-Rabin and Lucas tests, 218 minimal polynomial, 420 minimal, normal, and average order of Carmichael’s function λ, 193 Minkowski product, 104 modern computers, 24 modified q-Bernoulli numbers, 564 modified q-Bernoulli polynomials, 564 modified Stirling numbers, 500 modular group, 272 modular idempotent numbers, 431 Monica sets, 384 monoids with finite decomposition property, 148 multidimensional Smith’s determinants, 266 multiperfect numbers, 32, 35 multiplicative inverse of t, 250 multiplicative semigroup, 105 multiply perfect numbers, 32, 35, 71 Măobius categories, 148 Măobius function, 153, 154, 264, 268, 490, 496 Măobius function àG , 128 Măobius function associated with the dual poset, 157 Măobius function generated by the B-convolution, 116 Măobius function of n arguments, 109 Măobius function of P, 140 Măobius function of a direct product, 158 locally finit poset, 142 log-concave function, 524 log-concavity, 577 logarithm operator, 108 logarithmic and identric means, 486 logarithmic convexity, 471 logarithmic distribution function, 397 logarithmic Fourier series, 398 logarithmic polynomials and numbers, 533 logarithmically concave, 470 lovely numbers, 69 lower closed sets, 267 lower density, 236 Lubin-Tate groups, 566 Lucas sequence of the second kind, 240 Lucas sequence of the first kind, 240 Lucas sequences, 36 Lucas’ totient, 277 Mac Mahon’s Master theorem, 161 magic squares, 184 Magma system for computational algebra, 258 Maillet’s theorem, 207 Makowski-Schinzel conjecture, 235 Marriage theorem, 348 matched asymptotic expansions, 516 matrix-generated convolutions, 127 maximal generalization of Fermat’s theorem, 190 maximum and minimum exponents, 330 maximum of the exponents, 355 mean value, 401 meet semilattice, 267 meet-closed sets, 273 meet-closed subset, 268 meet-matrix, 268 630 INDEX Măobius function of a generalized integer, 125 Măobius function of a set of primes, 105 Măobius function of a trace monoid, 162 Măobius function of geometric lattices, 148 Măobius function on a subset of a power set, 154 Măobius functions of arbitrary direct factor sets, 135 Măobius inversion, 102104, 148, 490 Măobius inversion formula, 180, 251 Măobius inversions, 482 Măobius system, 104 Măobius type inversion theorems, 150 Măobius function, 184 Măobius function of an arithmetical semigroup, 148 Măobius-type function, 193 Măobius-type inversion formula, 135 Măobius-type inversion formulae, 125 noncototients, 208 nontotient, 230 nonunitary k-cycle, 76 nonunitary aliquot sequences, 76 nonunitary amicable pairs, 68 nonunitary divisors, 48 nonunitary perfect numbers, 48 nonunitary totient function, 287 normal order, 332 nowhere differentiable, 373, 375, 392, 400, 417 nowhere differrentiable, 399 number of decimal digits, 431 number of digit blocks of 11, 426 number of distinct prime divisors, 33, 110, 180 number of distinct prime factors, 28, 243, 250, 265, 283, 519 number of divisors, 35, 180, 264 number of odd digits, 425 number of solutions of the congruence, 279 number of square-free divisors, 130 number of terminating nines, 426 number of unitary divisors, 284 Nagells totient, 133, 278 Narkiewicz Măobius function, 144 Narkiewiecz Măobius function, 113 Narumi polynomials, 550 negative integer bases, 417 Newton’s formulae, 258 Nielsen polynomials, 467 nilpotent elements, 118, 290 Niven number, 381 no unitary m-superperfect numbers, 57 non-central Stirling numbers, 479 non-commuting lifting, 162 non-crossing set partitions, schuffle posets, 148 non-divisibility property, 202 occurrences of the block w, 425 odd amicable pairs, 64 odd perfect number, 25, 28, 29, 31, 33, 34, 41 odd perfect numbers, 19, 23, 26, 31 open problem, 203, 209, 222, 272, 349, 416 open problems, 31, 43, 76, 203 open question, 67, 209 operatorial theory, 469 optimization theory, 339 orbit, 490 orbits, 481 order of a modulo n, 189 631 INDEX positive definite, 273 power series, 100 power-harmonic number, 44 powerful, 52 practical numbers, 59, 359 preferential arrangements, 496 primality criterion, 254, 256 primary pseudoperfect number, 45 prime k-tuple conjecture, 218 prime factors of n (a, b), 255 prime factors of ϕ ( j) (n), 198 prime factors of the Euler numbers, 556 prime ideal, 293 prime ideal theorem, 160, 161 prime number theorem, 149, 150 prime-counting function π(n), 180 prime-power groups, 139 primitive character, 553 primitive divisors of Lucas sequences, 216 primitive element, 420 primitive root mod n, 251 primitive root of unity, 258 primitive roots, 189 primitive roots of unity, 138, 251 primitive solutions, 197, 198 primitive substitution, 374 probabilistic group theory, 364 product of all e-divisors, 58 product of all divisors, 55 product of digits, 383 product of divisors, 202 product set, 104 pseudoperfect numbers, 43 pseudoprime, 30 P´olya frequency sequence, 578 P´olya-Vinogradov inequality, 554 order of the group, 202 orthogonal projection, 394 orthogonality, 476, 477, 485 orthonormal basis, 185 oscillatory asymptotic, 524 other analogs of Lehmer’s problem, 215 overlap, 390 palindrome, 391 parallelotope, 420 parametric Măobius inversion, 145 Parrys α-expansion, 415 partial p, q-Stirling numbers, 486 partial ordering, 141 partial product, 128 partially ordered sets, 105 partition lattice q-analogs, 481 partitions, 461 Pascal’s triangle, 379 Pell sequence, 67 perfect number, 369 perfect totient number, 240 periodic mapping, 554 periodic perfect numbers, 17 periods of the Stirling numbers, 492 Phibonacci number, 223 Pillai arithmetic function, 182 Piltz divisor function, 106, 107 Pisot number, 415 Plato’s Republic, 17 pointwise product, 119 Poisson process, 341 poly-Bernoulli numbers, 501 polylogarithm function, 552 polynomial ring, 552 polynomial sequences, 402 poset, 139 poset-theoretic generalization of Smith’s theorem, 268 632 INDEX residue theorem, 478 resolution function, 367 resultant of cyclotomic polynomials, 262 reversing the digits, 429 Riemann hypothesis, 102, 129, 130 Riemann Hypothesis, 187 Riemann hypothesis, 262 Riemann zeta function, 99, 187, 263, 330, 510, 527 Riesz representation theorem, 185 right inversion, 123 right inversion formula of Măobius type, 123 rising factorial power, 461 roots of unity, 184, 393 Roth’s theorem, 426 Rudin-Schapiro sequence, 396 R´edei’s generalization of Euler’s theorem, 208 quadratic character, 554 quadratic field, 285, 293 quadratic fields, 32, 252 quadratic residue, 216 quadri-amicable numbers, 70 quasi-amicable pairs, 68 quasi-multiplicative functions, 270 quasi-multiplicativity, 116 quasi-orthogonality, 476 quasi-superperfect, 41 quasiperfect numbers, 36 queues, 464 Ramanujan expansions, 278 Ramanujan identities, 536 Ramanujan sum, 184, 258, 277 Ramanujan sums, 125, 137 Ramanujan sums on semigroups, 277 Ramanujan theorems, 544 Ramanujan’s sum, 265 ray method, 516 reciprocal GCD and LCM matrices, 272 recursive algorithm, 223 recursive set, 32 refined Stirling numbers, 487 regular S-representation, 285 regular arithmetic convolution, 144 regular convolution, 112 regular convolutions, 183 regular polygon, 188, 239 relatively k-prime, 136, 275 relatively prime amicable pairs, 65, 66 relatively prime amicable pairs of opposite parity, 66 remarkable identities, 282 representations of real numbers, 432 repunit number, 383 residue classes, 387 residue classes modulo n, 249 saddle point method, 516, 583 Salem numbers, 415 Schemmel totient, 184 Schemmel totient function, 229 Schemmel’s totient, 133, 216, 276 Schemmel-Nagell totient, 133, 276 Schinzel’s H -Hypothesis, 380 Schinzel’s conjecture, 218 Schinzel’s Hypothesis H, 198, 221 Schinzel-Szekeres function, 356 Schmidt’s subspace theorem, 418, 508 Schrăodinger operators, 391 Schurs theorem, 260 secant numbers, 559 second order Eulerian numbers, 580 Selberg’s identity, 160 Selberg’s upper bound sieve, 226 self-number, 384 semi-multiplicativity, 116 633 INDEX semi-unitary divisor, 281 semi-unitary greatest common divisor, 271 semi-unitary product, 128 semigroup approach to the Fermat and Euler’s theorems, 191 semigroups with unique factorization, 150 semisimple rings, 333 sequence (Bk (mod n)), 555 sequence of Eulerian numbers, 577 sequences of Fibonacci, Pell, and Lucas, 59 set of totients, 206 Shapiro-Pillai function, 200 Shintani zeta functions, 536 Siegel-Walfisz theorem, 248 sieve of Eratosthenes, 338 Sieve theory, 103 sieve-theoretical problems, 355 sigmoid function, 569 signed Eulerian polynomial, 570 signless q-Stirling numbers of the first kind, 481 signless Bernoulli numbers, 525 Silverman constant, 182 simple groups, 157 simple pole, 415 Sita Ramaiahs Măobius function, 113 small prime divisors, 341 small sieve, 359 smallest prime totative, 248 Smarandache function, 22, 210, 237 Smith brothers, 384 Smith number, 383 Smith-Jones number, 384 sociable number of order k, 75 sociable numbers, 72 soluble group, 155, 157 soluble groups, 154 special cyclotomic polynomials, 252 spectral norm of a matrix, 274 square-reduced residue system, 289 square-totient of R Sivaramakrishnan, 289 squarefree integers, 33, 207, 265 squarefree number, 35, 46 squarefree numbers, 136 squarefull, 209 squarefull integers, 331 staggered sums, 429 statistic on permutations, 479 statistical convergence, 331 statistical limit, 335 Stevens totient, 280 Stevens’ totient, 280 Stieltjes transforms, 525 Stirling functions of the first kind, 473 Stirling functions of the second kind, 473 Stirling numbers associated to a matrix, 475 Stirling numbers of complex argument, 487 Stirling numbers of complex arguments, 523 Stirling numbers of fractional order, 472 Stirling numbers of the first kind, 462, 479 Stirling numbers of the first kind st (n, k) associated to the sequence t, 474 Stirling numbers of the second kind, 460, 578 Stirling numbers of the third kind, 464 Stirling polynomials, 466, 469 Strassen functional, 343 634 INDEX strong log-concavity, 508 strong regular A-convolution, 119 strongly q-additive, 411 strongly q-log-concave, 514 subadditive functions, 344 subblock occurrences, 422 subgroup, 151, 157 sublime numbers, 22 submatrix, 269 subring, 142 subspace, 185 sugcd-closed, 271 sum of jth powers of totatives, 242 sum of digits, 371, 499, 551 sum of digits of primes, 379 sum of digits with prime indices, 390 sum of divisors, 180, 264 sum of prime digits, 390 sum of totatives, 242 sum-product numbers, 383 super-lovely pair, 69 super-multiplicative, 355 supermultiplicative, 346 superperfect number, 58 superperfect numbers, 23, 38, 69 Suzanne sets, 384 Sylow p-subgroup, 155 Sylow subgroups, 158 symmetric signed digit expansion, 422 symmetric system, 422 symmetries and dualities, 476 the characteristic, 511 The exponential A-convolution, 118 theorem by J Touchard, 29 theorem by Touchard, 35 theorem of Adams, 541 theorem of Howard, 551 theorem of Sylvester, 25 theorem of Sylvester and Lipschitz, 548 theoretical physics, 375 theory of Măobius function, 139 Thue-Morse sequence, 379 Thue-Morse sequence, 390 total number of prime divisors, 180 total number of prime factors, 26, 133, 137 totatives, 179, 242 totient functions, 125 totient-free residue classes, 206 Touchard’s congruence, 495 trace languages, 161 trace monoids, 161 transcendence, 391 transcendental, 426 transitive orientation, 162 triangular matrices, 265 triperfect numbers, 34 twin dragon, 421 twin primes, 234 twin-prime constant, 219 tangent numbers, 527, 559, 577 Taylor expansion, 488 te Rieles trick, 63 Teichmăuller character, 554 Thabits rule, 17, 61 Thacker’s function, 243 Thacker’s totient function, 289 ud-closed, 271 ultrametric Banach space, 546 unambiguous lifting, 162 unambiguous Măobius function, 162 unequal characters, 554 uniform asymptotic, 524 635 INDEX unitary hyperperfect numbers, 51 unitary multiply superperfect numbers, 48 unitary Măobius function, 111, 272 unitary perfect numbers, 45 unitary product, 110 unitary product set, 111 unitary quasi superperfect numbers, 47 unitary quasi-sociable numbers, 75 unitary quasiperfect numbers, 47 unitary Smith determinants, 271 unitary sociable numbers, 74 unitary superperfect numbers, 47 unitary totient function, 213, 266 universal Bernoulli numbers, 552 universal generated, 385 unsigned Lah numbers, 464 unsigned Stirling numbers of the first kind, 462 unsolved problems on generalized integers, 125 upper density, 345 uniform asymptotic expansion, 518 uniform limit distribution mod m, 389 uniform probability measure, 343 uniform summability, 413 uniformly distributed mod m, 389 uniformly distributed mod 1, 410 uniformly distributed modulo 1, 397 uniformly periodic, 427 unimodal, 577 unique factorization domains, 32 unitary m-perfect numbers, 57 unitary abundant numbers, 46 unitary aliquot sequences, 74 unitary almost perfect numbers, 47 unitary almost superperfect number, 47 unitary almost-sociable numbers, 75 unitary amicable numbers, 67 unitary analogue of Euler’s totient, 242 unitary analogue of Legendre’s totient, 284 unitary analogue of Nagell’s totient, 281 unitary analogue of Ramanujan’s sum, 267 unitary analogue of Schemmel’s totient, 281 unitary analogue of Smith’s determinant, 266 unitary analogue of the Carmichael’s conjecture, 229 unitary analogues of generalized Ramanujan sums, 277 unitary convolution, 112, 127 unitary divisor, 46 unitary harmonic numbers, 49, 50 valence function of ϕ, 226 vector space, 145, 286, 420, 481 von Mangoldt function, 160, 259 von Staudt-Clausen and Voronoi type congruences, 554 von Staudt-Clausen theorem, 245, 539 von Sterneck and the Cohen totients, 275 Waring’s problem, 421 weak convergence of distribution functions, 343 weakly-composite, 340 weird numbers, 43, 44 well-distributed mod 1, 411 636 INDEX Worpitzky type formulas, 572 wreath products, 490 Weyl derivative, 474 Weyl’s lemma, 421 Wieferich prime, 212 Wieferich primes, 540 Wiegandt convolution, 139 Wiener measure, 344 Wilson quotient, 540, 547 Wilson theorem, 574 Wilson’s theorem, 191, 488, 542 Worpitzky identity, 568 Zeckendorf sum-of-digits function, 398 Zeckendorf Thue-Morse sequence, 399 zero divisors, 118 Zsigmondy-Birkhoff-Vandiver theorem, 202 колхоз 11/14/06 637 ... infinitary, exponential totient sum of divisors function number of divisors function number of distinct, resp total number, of prime factors of n sum of kth powers of divisors of n unitary, bi-unitary,... 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... Stirling numbers Carlitz’ degenerate Stirling numbers Dickson-Stirling numbers; resp Hsu-Shiue-Stirling numbers associated Stirling numbers q-Stirling numbers signless q-Stirling numbers of the