Sandor J., Crstici B. Handbook of Number Theory Volume 2

Kluwer Academic Publishers, 2004. 635 p. ISBN: 1-4020-2546-7 (HB)
ISBN: 1-4020-2547-5 (e-book)Basic Symbols
Basic Notations
Perfect Numbers: Old and New Issues; Perspectives
ntroduction
Some historical facts
Even perfect numbers
Odd perfect numbers
Perfect, multiperfect and multiply perfect numbers
Quasiperfect, almost perfect, and pseudoperfect numbers
Superperfect and related numbers
Pseudoperfect, weird and harmonic numbers
Unitary, bi-unitary, infinitary-perfect and related numbers
Hyperperfect, exponentially perfect, integer-perfect and γ -perfect numbers
Multiplicatively perfect numbers
Practical numbers
Amicable numbers
Sociable numbersGeneralizations and Extensions of the M ¨Obius FunctionM¨obius functions generated by arithmetical products (or convolutions)
M¨obius functions defined by Dirichlet products
Unitary M¨obius functions
Bi-unitary M ¨obius function
M¨obius functions generated by regular convolutions
K -convolutions and M ¨obius functions. B convolution
Exponential M¨obius functions
l.c.m.-product (von Sterneck-Lehmer)
Golomb-Guerin convolution and M¨obius function
max-product (Lehmer-Buschman)
nfinitary convolution and M ¨obius function
M¨obius function of generalized (Beurling) integers
Lucas-Carlitz (l-c) product and M ¨obius functions
Matrix-generated convolution
M¨obius function generalizations by other number theoretical considerations
Apostol’s M¨obius function of orderk
Sastry’s M ¨obius function
M¨obius functions of Hanumanthachari and Subrahmanyasastri
Cohen’s M ¨obius functions and totients
Klee’s M ¨obius function and totient
M¨obius functions of Subbarao and Harris; Tanaka
and Venkataraman and Sivaramakrishnan
M¨obius functions as coefficients of the cyclotomic polynomial
M¨obius functions of posets and lattices
Introduction, basic results
Factorable incidence functions, applications
nv ersion theorems and applications
M¨obius functions on Eulerian posets
Miscellaneous results
M¨obius functions of arithmetical semigroups, free groups, finite groups, algebraic number fields, and trace monoids
M¨obius functions of arithmetical semigroups
Fee abelian groups and M ¨obius functions
M¨obius functions of finite groups M¨obius functions of algebraic number and
function-fields
Trace monoids and M¨obius functionsThe Many Facets of Euler’s TotientThe 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
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
Euler minimum function
Lehmer’s conjecture, generalizations and extensions
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 involvingϕ and other arithmetic functions
The composition ofϕ and other arithmetic functions
Perfect totient numbers and related results 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 withϕ
Smith’s determinant
Poset-theoretic generalizations
Factor-closed, gcd-closed, lcm-closed sets, and
related determinants
nequalities
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
Nonunitary, infinitary, exponential-totients
Thacker’s, Leudesdorf’s, Lehmer’s, Golubev’s totients. Square totient, core-reduced totient, M-void totient,
additive totient
Euler totients of arithmetical semigroups, finite groups, algebraic number fields, semigroups, finite commutative rings, finite Dedekind domainsSpecial Arithmetic Functions Connected with the Divisors, or with the Digits of a NumberSpecial 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
Hooley’s function
Extensions of the Erd ¨os conjecture (theorem)
The divisors in residue classes and in intervals
Divisor density and distribution (mod 1) on divisors
The fractal structure of divisors
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
Niv en 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
Uniform - and well - distributions of α s q (n )
The G -ary digital expansion of a number
The sum-of-digits function for negative integer bases
The sum-of-digits function in algebraic number fields
The symmetric signed digital expansion
Infinite sums and products involving the sum-of-digits function
Miscellaneous results on digital expansionsStirling, Bell, Bernoulli, Euler and Eulerian NumbersStirling and Bell numbers
Stirling numbers of both kinds, Lah numbers
dentities for Stirling numbers
Generalized Stirling numbers
Congruences for Stirling and Bell numbers
Diophantine results
Inequalities and estimates
Bernoulli and Euler numbers
Definitions, basic properties of Bernoulli numbers and polynomials
dentities
Congruences for Bernoulli numbers and polynomials. Eulerian numbers and polynomials
Estimates and inequalities