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

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.