A generalization of the infinitary divisibility relation: Algebraic and analytic properties

Abstract

We consider a generalized type of unique factorization of the positive integers with restrictions on the exponents and view them as a family of arithmetic convolutions and divisibility relations, similar to the convolutions defined by Narkewicz [On a class of arithmetical convolutions, Colloq. Math. 10 (1963) 81–94]. We introduce special types of multiplicativity corresponding to these convolutions, and discuss algebraic properties of the associated arithmetic convolutions and analogs of the Möbius functions. We also prove asymptotics for analogs of the totient function, totient summatory function, and divisor summatory function.

On a General Class of A-functions

“…Other generalizations include the generalizations of the infinitary divisibility relation of Cohen (see [5,6,9,10,21]), which will be actively discussed in this work, and the generalization of convolutions to convolutions involving weight functions (see [11,12]).…”

“…On the other hand, we call A a cross-convolution if it is multiplicative but not homogeneous. The concept of cross-convolution in the context of A-functions is due to Tóth [30], and was formalized and generalized in [5]. If we let {A λ } λ∈Λ be an indexed set of homogeneous A-functions and let f ∶ P → Λ be a function, then we may define A f to be the multiplicative A-function such that A f (p a ) = A f (p) (p a ) for all p ∈ P and all a ∈ N, with A f (1) = {1} by convention.…”

“…The original cross-convolution A-functions of Tóth [30], the structured A-functions of [5], and the mixed-multiplicative A-functions of Litsyn and Shevelev [21] are examples of families of cross-convolution A-functions appearing in the literature.…”

“…Taking inspiration from the final family of A-functions, the k-ary A-functions of Cohen, the authors of [5] introduced the following definition for what they called the iterate of an A-function:…”

“…Remark 1. In using the notation A(d) ∩ A n d , the authors of [5] have circumvented an awkward process present in much of the literature, whereby in order to define a new family of A-functions it was necessary to first define an analogue of the GCD-function and express which divisors d of n were to be included in the set A(n) on the basis of some condition on this analogue of the GCD-function. (cf.…”