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.

Key words： Number of subgroups; asymptotic formula; error term; exponential sums

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