On factorizations into coprime parts

TL;DR

This paper investigates the asymptotic behavior of sums involving factorizations of integers into coprime parts, extending known results to various powers and providing bounds for these sums.

Contribution

It establishes new asymptotic bounds for sums of factorizations into coprime parts for all real powers, including negative powers, advancing understanding of factorization functions.

Findings

Derived asymptotic bounds for sums of coprime factorizations for all real 

Extended results to negative powers of factorization counts

Provided bounds for sums involving general factorization functions

Abstract

Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the sums of $F(n)^{\beta}$ and $G(n)^{\beta}$ up to $x$ for all real $\beta$ and the asymptotic bounds for $f(n)^{\beta}$ and $g(n)^{\beta}$ for all negative $\beta$.