Error term concerning number of subgroups of group ${\text{ℤ}}_m\times {\text{ℤ}}_n$ with $m^2+n^2\le x$

Abstract

Let ${\text{ℤ}}_m$ be the additive group of residue classes modulo m. Let s(m, n) denote the number of subgroups of the group ${\text{ℤ}}_m\times {\text{ℤ}}_n$, where m and n are arbitrary positive integers. For any $x\ge 1$, we consider the asymptotic behavior of $D_s\left(x\right):=\sum m^2+n^2\le xS(M,n)$ and obtain an asymptotic formula by using the elementary method.

Keywords

Number of subgroups / asymptotic formula / error term / exponential sums

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