Gaussian integers partition in power-free numbers product

Abstract

Let $\mathrm{g}_1 (\alpha)$ be the number of Gaussian integer $\alpha$ representation in a product of square-free factors. Let $\mathrm{g}_2 (\alpha)$ be the number of Gaussian integer $\alpha$ representation in a product of power-free factors. In this paper we consider their summatory functions $\sum_{N(\alpha)\le x} \mathrm{g}_1 (\alpha)$ and $\sum_{N(\alpha)\le x} \mathrm{g}_2 (\alpha)$ and obtain asymptotic formulas for them. Also, we prove analogue of K\'{a}tai-Subbarao theorem to study the distribution of $\mathrm{g}_2 (\alpha)$ in increasing norm order case.