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

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

Categories: math.NT