On the average number of subgroups of the group ${\mathbb{Z}}_{n_1}\times {\mathbb{Z}}_{n_2}$

Abstract

Let ${\mathbb{Z}}_n$ be the additive group of residue classes modulo $n.$ For any positive integers $n_1$ and $n_2$, let $s(n_1,n_2)$ and $c(n_1,n_2)$ denote the number of subgroups and the number of cyclic subgroups of the group ${\mathbb{Z}}_{n_1}\times {\mathbb{Z}}_{n_2}$, respectively. The aim of this paper is to study the asymptotic behavior of the sums ${\sum }_{n_1,n_2\le x}s(n_1,n_2)$ and ${\sum }_{n_1,n_2\le x}c(n_1,n_2)$. Some sharper asymptotic results are given for the these sums. Mean values of the error terms are also studied.