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

Abstract

The k-ary divisibility relations are a class of recursively defined relations beginning with standard divisibility and culminating in the so-called infinitary divisibility relation. We examine the summatory functions corresponding to the k-ary analogues of various popular functions in number theory, proving various results about the structure of the k-ary divisibility relations along the way.

On a General Class of A-functions

“…Subsequently, various works were produced which studied and generalized the unitary divisibility relation and associated convolution (see [8,16,19,31]). In several cases, entire classes of A-functions were introduced, viewing the usual divisibility relation (which corresponds to the A-function D) and the unitary divisibility relation as the first two elements of a sequence of A-functions that preserved some property or theme (see [1,3,9,10,17,28]).…”