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Home , Aliquot sequence, Amicable numbers, Arithmetic function, Bell number, Carmichael function, Dedekind number

Index

(A, r)-powerful, 114 k-Eulerian posets, 148 (m, k)-perfect, 41, 42 k-free part, 134 (α, β)-multiamicable , 71 k-full, 236 3x + 1 problem, 190 k-full part, 134 A-analogue of Smith’s k-harmonic , 44 , 266 k-hyperperfect numbers, 50 A-, 112, 278 k-Mobius¨ of a poset, 147 A-multiplicative, 114 k-multiperfect numbers, 30 A-rational function, 114 k-, 32 Ak-, 113 k-perfect numbers, 33 B-convolution, 116 k-reduced residue system, 276 B-splines, 471 k-, 383 b-additive, 404 k-th power of an arithmetic e-multiperfect, 52 function, 108 e-perfect numbers, 52 k-, 46 e-superperfect numbers, 53 kth cyclotomic , 561 f -expansions, 432 kth iterate of f , 196 f -generated number, 386 l p norm (p ≥ 1) of matrices, 274 f -iterative , 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 , 136 n-dimensional symplex, 568

619 INDEX n- with η-colored q-, 286 head, 480 q-, 561 n-th prime pn, 180 q-Euler , 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 , 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-, 155 q-tangent numbers, 559 p-local integers, 553 q-trigonometric function, 561 − p-Stirling numbers, 472 q L-, 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 , 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 , 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-, 562 155 q-binomial coefficient, 471 abelian number fields, 553 q-binomial coefficients, 480, 506 abstract theorems, 154

620 INDEX abundant numbers, 17, 43 , 582 action of the , 490 and Mellin-Perron 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 , 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 field, 159, 263, 419 arithmetic progressions, 136, 151, algebraic number fields, 193 220, 229, 254 algebraic , 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 , 72, 74 associated Stirling numbers, 502, 580 all-Niven, 383 , 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 , 36, 60, 63 asymptotic expansions of Stirling amicable numbers in special numbers of the second kind, , 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 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 , 461 Bohr-Fourier spectrum, 413 Bell numbers, 493 Bolyai-Renyi´ expansions, 432 , 484, 495 Bombieri’s sieve, 227 Bell sequence, 512 Boole summation formula, 577 Benford law, 430 Boolean , 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 , 461 britannic number system, 407 Bernoulli numbers, 243, 244, 495, Brownian motion, 344 500, 518, 525 Brun’s sieve, 235 Bernoulli numbers of fractional , 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-, 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 , 378 congruence properties, 477 character group, 151 congruence properties of σ (m)(n),42 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 on words, 390 conjugate Bernoulli polynomials, 538 common generalization of Liouville’s consecutive , 342 function and the Mobius¨ consecutive prime factors, 337 function, 137 constants, 181 commutative group, 106 continued fractions, 432, 537, 561 commutative , 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 , 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 , 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, , 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 , 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 , 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 ’s ”Elements”, 16 exponential convolution, 117 Euclid-Euler theorem, 21 exponential cycles, 75 , 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 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 methods of integers, 256 numbers, 540 factorization of Bernoulli numerators, Frattini chief factor, 155 556 free , 151 falling factorial, 478 frequency of a block, 421 falling factorial power, 460 friendly pair, 70 , 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, , 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 , 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 , 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 , 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 , 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 , 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 , 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 , 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 , 364 iterates, 75 lcm-closed sets S, 271 iteration, 42 , 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 , 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-, 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 , 21, 22, 234 log-concavity, 577 Mersenne primes, 15, 19, 41, 42, 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 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 , 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 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 , 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 , 258 Phibonacci number, 223 primitive roots, 189 Pillai arithmetic function, 182 primitive roots of unity, 138, 251 Piltz , 106, 107 primitive solutions, 197, 198 Pisot number, 415 primitive substitution, 374 Plato’s Republic, 17 probabilistic , 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 , 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, , 216 262 quadri-amicable numbers, 70 reversing the digits, 429 quasi-amicable pairs, 68 , 102, 129, 130 quasi-multiplicative functions, 270 Riemann Hypothesis, 187 quasi-multiplicativity, 116 Riemann hypothesis, 262 quasi-orthogonality, 476 , 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 , 66 secant numbers, 559 remarkable identities, 282 second order Eulerian numbers, 580 representations of real numbers, 432 Selberg’s identity, 160 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 , 338 Stieltjes transforms, 525 , 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 of order k,75 Stirling numbers of the third kind, 464 sociable numbers, 72 , 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 , 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 , 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 , 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 , 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