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

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