On the error term concerning the number of subgroups of the groups $\mathbb {Z}_m \times \mathbb {Z}_n$ with $m,n\le x$
Volume 183 / 2018
Acta Arithmetica 183 (2018), 285-299
MSC: Primary 11A25, 11N37; Secondary 20K01, 20K27.
DOI: 10.4064/aa171111-7-2
Published online: 23 March 2018
Abstract
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 $n$ are arbitrary positive integers. W. G. Nowak and L. Tóth (2014) investigated the asymptotic behavior of the sums $\sum_{m,n\leq x} s(m,n)$ and $\sum_{m,n\leq x} c(m,n)$. We improve the error terms of these asymptotic formulas.
