On the weighted average number of subgroups of ${\mathbb {Z}}_{m}\times {\mathbb {Z}}_{n}$ with $mn\leq x$

TL;DR

This paper investigates the asymptotic behavior of weighted sums of the number of subgroups and cyclic subgroups of the direct product of residue class groups, providing insights into their growth as the product of group orders increases.

Contribution

It introduces asymptotic formulas for weighted sums of subgroup counts in direct product groups, extending understanding of their distribution and growth.

Findings

Asymptotic formulas derived for _s(x) and _c(x) functions

Quantitative growth rates of subgroup counts established

Enhanced understanding of subgroup distribution in product groups

Abstract

Let $\mathbb{Z}_{m}$ be the additive group of residue classes modulo $m$. For any positive integers $m$ and $n$, let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups and cyclic subgroups of the group ${\mathbb{Z}}_{m}\times {\mathbb{Z}}_{n}$, respectively. Define $$ \widetilde{D}_{s}(x) = \sum_{mn\leq x}s(m,n)\log\frac{x}{mn} \quad \quad \widetilde{D}_{c}(x) = \sum_{mn\leq x}c(m,n)\log\frac{x}{mn}. $$ In this paper, we study the asymptotic behaviour of functions $\widetilde{D}_{s}(x)$ and $\widetilde{D}_{c}(x)$.