Index (A, r)-powerful, 114 k-Eulerian posets, 148 (m, k)-perfect, 41, 42 k-free part, 134 (α, β)-multiamicable numbers, 71 k-full, 236 3x + 1 problem, 190 k-full part, 134 A-analogue of Smith’s k-harmonic number, 44 determinant, 266 k-hyperperfect numbers, 50 A-convolution, 112, 278 k-Mobius¨ function of a poset, 147 A-multiplicative, 114 k-multiperfect numbers, 30 A-rational function, 114 k-perfect number, 32 Ak-convolutions, 113 k-perfect numbers, 33 B-convolution, 116 k-reduced residue system, 276 B-splines, 471 k-Smith number, 383 b-additive, 404 k-th power of an arithmetic e-multiperfect, 52 function, 108 e-perfect numbers, 52 k-unitary perfect number, 46 e-superperfect numbers, 53 kth cyclotomic polynomial, 561 f -expansions, 432 kth iterate of f , 196 f -generated number, 386 l p norm (p ≥ 1) of matrices, 274 f -iterative sequence, 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 divisor, 136 n-dimensional symplex, 568 619 INDEX n-permutations with η-colored q-Euler function, 286 head, 480 q-Euler numbers, 561 n-th prime pn, 180 q-Euler polynomials, 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 integers, 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-group, 155 q-tangent numbers, 559 p-local integers, 553 q-trigonometric function, 561 − p-Stirling numbers, 472 q L-series, 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 integer, 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 power series, 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-Bernoulli polynomials, 562 155 q-binomial coefficient, 471 abelian number fields, 553 q-binomial coefficients, 480, 506 abstract prime number theorems, 154 620 INDEX abundant numbers, 17, 43 analytic continuation, 582 action of the cyclic group, 490 and Mellin-Perron summation 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 arithmetic function, 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 algebraic number field, 159, 263, 419 arithmetic progressions, 136, 151, algebraic number fields, 193 220, 229, 254 algebraic number theory, 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 aliquot sequence, 72, 74 associated Stirling numbers, 502, 580 all-Niven, 383 asymptotic analysis, 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 amicable numbers, 36, 60, 63 asymptotic expansions of Stirling amicable numbers in special numbers of the second kind, sequences, 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 unit 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 Bell number, 461 Bohr-Fourier spectrum, 413 Bell numbers, 493 Bolyai-Renyi´ expansions, 432 Bell polynomials, 484, 495 Bombieri’s sieve, 227 Bell sequence, 512 Boole summation formula, 577 Benford law, 430 Boolean lattice, 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 Bernoulli number, 461 britannic number system, 407 Bernoulli numbers, 243, 244, 495, Brownian motion, 344 500, 518, 525 Brun’s sieve, 235 Bernoulli numbers of fractional order, 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-unitary divisor, 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 multiplication, 378 congruence properties, 477 character group, 151 congruence properties of σ (m)(n),42 characteristic 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 combinatorics on words, 390 conjugate Bernoulli polynomials, 538 common generalization of Liouville’s consecutive divisors, 342 function and the Mobius¨ consecutive prime factors, 337 function, 137 constants, 181 commutative group, 106 continued fractions, 432, 537, 561 commutative ring, 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