On the weighted average number of subgroups of \mathbb Z_m× \mathbb Z_n with mn≤ x

2020/08/18 by Kiuchi, Isao, Eddin, Sumaia Saad
#11A25 #11N37 #11Y60 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2008.07850

Abstract

Let ℤ_m be the additive group of residue classes modulo m. For any positive integers m and n, let s(m,n) and c(m,n) denote the total number of subgroups and cyclic subgroups of the group ℤ_m× ℤ_n, respectively. Define \widetildeD_s(x) = ∑_{mn≤ x}s(m,n)log(x)/(mn) \widetildeD_c(x) = ∑_{mn≤ x}c(m,n)log(x)/(mn). In this paper, we study the asymptotic behaviour of functions \widetildeD_s(x) and \widetildeD_c(x).

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