Asymptotics of the counting function of $k$-th power-free elements in an arithmetic semigroup

TL;DR

This paper derives the asymptotic behavior of the counting function for k-th power-free elements within an additive arithmetic semigroup exhibiting exponential prime counting growth, extending previous work to finite k.

Contribution

It provides new asymptotic formulas for k-th power-free elements in additive semigroups with exponential prime growth, generalizing earlier results for infinite k.

Findings

Asymptotic formulas for counting functions derived

Extended previous results to finite k

Applicable to semigroups with exponential prime growth

Abstract

For any $k>1$, we find the asymptotics of the counting function of $k$-th power-free elements in an additive arithmetic semigroup with exponential growth of the abstract prime counting function. This paper continues the authors' earlier research dealing with the case of infinite $k$.