On a class of arithmetic convolutions involving arbitrary sets of integers

Abstract

Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum is extended over the $S$-divisors of $n$. We determine the sets $S$ such that the $S$-convolution is associative and preserves the multiplicativity of functions, respectively, and discuss other basic properties of it. We give asymptotic formulae with error terms for the functions ${\sigma }_S(n)$ and ${\tau }_S(n)$, representing the sum and the number of $S$-divisors of $n$, respectively, for an arbitrary $S$. We improve the remainder terms of these formulae and find the maximal orders of ${\sigma }_S(n)$ and ${\tau }_S(n)$ assuming additional properties of $S$. These results generalize, unify and sharpen previous ones. We also pose some problems concerning these topics.