  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 $\widetilde{D}_{c}(x)$.
