Arithmetic functions associated with infinitary divisors of an integer

Abstract

The infinitary divisors of a natural number n are the products of its divisors of the form pyα2α, where py is a prime-power component of n and ∑αyα2α (where yα=0 or 1) is the binary representation of y. In this paper, we investigate the infinitary analogues of such familiar number theoretic functions as the divisor sum function, Euler's phi function and the Möbius function

Arithmetical Functions Associated with the k-ary Divisors of an Integer

“…Let ∞ ( ) be the set of infinitary divisors of introduced and studied by Cohen [3,7]. The infinitary divisibility relation can be thought of as the end behavior of the recursion defining the -ary divisibility relations.…”

“…We will employ techniques already used in [7,8] to derive the result for the infinitary and unitary cases, respectively. Definition 5.…”

Menon’s identity with respect to a generalized divisibility relation

“…The numbers q∈S q θq , 0 ≤ θ q ≤ λ q , are referred to as the exponentially b-ary divisors of n in [9]. Thus the exponentially b-ary divisibility relation is a divisibility relation of type f , where for all primes p, f p (a) is the smallest power b j , j ∈ Z + ∪ {0}, for which a j is nonzero in the b-ary representation a 0 + a 1 b + a 2 b 2 + · · · of a. Cohen and Hagis ( [2]) as well as Vasu and Subrahmanya Sastri ( [8]) study the exponentially binary divisibility relation. Ramamoorthi, Vasu and Subrahmanya Sastri ( [7]) study the exponentially ternary divisibility relation.…”

A modified Möbius μ-function

“…Moreover, if f is a non-zero multiplicative function, then their inverses under each convolutions are also multiplicative, cf. [2,4,5].…”

“…Besides µ ∞ Cohen and Hagis [2] also investigated the functions τ ∞ (n) and σ ∞ (n), denoting the number and the sum of the infinitary divisors of n, proving asymptotic formulae for the summatory functions of τ ∞ (n) and σ ∞ (n). Asymptotic formulae for the corresponding bi-unitary functions τ * * (n) and σ * * (n) were established in papers [11,12,13].…”