On the weighted average number of subgroups of ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$ with $mn\le x$

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 $${\tilde{D}}_s(x)=\sum _{mn\le x}s(m,n)\log \frac{x}{mn}{\tilde{D}}_c(x)=\sum _{mn\le x}c(m,n)\log \frac{x}{mn}.$$ In this paper, we study the asymptotic behaviour of functions ${\tilde{D}}_s(x)$ and ${\tilde{D}}_c(x)$.

9 pages