On the error term concerning the number of subgroups of the groups $\mathbb {Z}_m \times \mathbb {Z}_n$ with $m,n\le x$
László Fejes Tóth, Wenguang Zhai · Acta Arithmetica · 2018
Let $\mathbb{Z}_{m}$ be the additive group of residue classes modulo $m$. Let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups and the number of cyclic subgroups of the group $\mathbb{Z}_{m}\times \mathbb{Z}_{n}$, respectively, where $m$ and $n$