On the number of subgroups of the group $\mathbb{Z}_{m_{1}} \times \mathbb{Z}_{m_{2}}$ with $m_{1}m_{2}\leq x$ such that $m_{1}m_{2}$ is a $k$-th power

Yankun Sui; Dan Liu; Boling Zhou

paper

On the number of subgroups of the group $Z_{m_{1}} \times Z_{m_{2}}$ with $m_{1} m_{2} \leq x$ such that $m_{1} m_{2}$ is a $k$ -th power

arXiv:2504.00331

Abstract

Let $Z_{m}$ be the additive group of residue classes modulo $m$ and $s (m_{1}, m_{2})$ denote the number of subgroups of the group $Z_{m_{1}} \times Z_{m_{2}}$ , where $m_{1}$ and $m_{2}$ are arbitrary positive integers. We consider sums of type $m_{1} m_{2} \leq x m_{1} m_{2} \in N_{k} \sum s (m_{1}, m_{2})$ , where $N_{k}$ is the set of $k$ -th power of natural numbers. In particular, we deduce asymptotic formulas with $k = 2$ and $k = 3$ .

16 pages

On the number of subgroups of the group Zm1​​×Zm2​​ with m1​m2​≤x such that m1​m2​ is a k-th power

Abstract

On the number of subgroups of the group $Z_{m_{1}} \times Z_{m_{2}}$ with $m_{1} m_{2} \leq x$ such that $m_{1} m_{2}$ is a $k$ -th power