Menon's identity and arithmetical sums representing functions of several variables

TL;DR

This paper generalizes Menon's identity to sums involving multiple variables and applies it to count cyclic subgroups in direct products of cyclic groups, also exploring extensions of Menon's identity not previously documented.

Contribution

It introduces a multi-variable generalization of Menon's identity and provides new formulas for counting cyclic subgroups in complex group structures.

Findings

Derived a formula for cyclic subgroups in direct products of cyclic groups

Extended Menon's identity to multi-variable sums

Identified new one-variable extensions of Menon's identity

Abstract

We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary orders. We also point out extensions of Menon's identity in the one variable case, which seems to not appear in the literature.