Mean values of multivariable multiplicative functions and applications to the average number of cyclic subgroups and multivariable averages associated with the LCM function

TL;DR

This paper employs multiple zeta functions to derive precise asymptotic formulas for averages of multivariable multiplicative functions, with applications to cyclic subgroup counts and LCM-related averages.

Contribution

It introduces a method using multiple zeta functions to analyze multivariable multiplicative functions and proves conjectures on cyclic subgroup averages and LCM-based multivariable averages.

Findings

Asymptotic formulas for multivariable multiplicative functions derived.

Confirmed conjectures on the average number of cyclic subgroups.

Established new multivariable averages related to the LCM function.

Abstract

We use multiple zeta functions to prove, under suitable assumptions, precise asymptotic formulas for the averages of multivariable multiplicative functions. As applications, we prove some conjectures on the average number of cyclic subgroups of the group ${\mathbb Z}_{m_1}\times\dots\times {\mathbb Z}_{m_n}$ and multivariable averages associated with the LCM function.