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 Tóth; Wenguang Zhai

doi:10.4064/aa171111-7-2

Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

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

László Tóth, Wenguang Zhai 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.

Authors

László TóthDepartment of Mathematics
University of Pécs
Ifjúság útja 6
H-7624 Pécs, Hungary
e-mail
Wenguang ZhaiDepartment of Mathematics
China University of Mining and Technology
Beijing 100083, P.R. China
e-mail
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Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Acta Arithmetica

On the error term concerning the number of subgroups of the groups \mathbb {Z}_m \times \mathbb {Z}_n with m,n\le xm,n\le x

Volume 183 / 2018

Abstract

Authors

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On the error term concerning the number of subgroups of the groups $\mathbb {Z}_m \times \mathbb {Z}_n$ with $m,n\le x$