On the error term concerning the number of cyclic subgroups of Z_l \times Z_m \times Z_n with lmn\leqslant x

TL;DR

This paper studies the asymptotic behavior of the sum of cyclic subgroups in the direct product of three cyclic groups, providing an asymptotic formula and bounds on the error term using complex analysis techniques.

Contribution

It introduces an asymptotic formula for the sum of cyclic subgroups in Z_l × Z_m × Z_n and bounds the error term's mean-square, advancing understanding of subgroup enumeration.

Findings

Derived an asymptotic formula for D3c(x)

Established an upper bound for the error term's mean-square

Applied complex analysis methods to subgroup counting

Abstract

Let Zn denote the additive group of residue classes modulo n. Let c(l,m,n) denote the number of cyclic subgroups of Zl *Zm *Zn. For any x > 1, we consider the asymptotic behavior of D3c(x):= \sum_{lmn\leq x} c(l,m,n), obtain an asymptotic formula by complex method, and get an upper bound for the integral mean-square of the error term in that asymptotic formula.