A Menon-type Identity derived using Cohen-Ramanujan sum

TL;DR

This paper presents a new derivation of a Menon-type identity involving generalized gcd sums and Cohen-Ramanujan sums, expanding the mathematical understanding of gcd-related identities.

Contribution

It introduces an alternative method to derive a Menon-type identity using Cohen-Ramanujan sums, complementing previous approaches.

Findings

Derived a Menon-type identity using Cohen-Ramanujan sums.

Provided an alternative proof method for gcd sum identities.

Enhanced understanding of generalized gcd functions and their properties.

Abstract

Menon's identity is a classical identity involving gcd sums and the Euler totient function $\phi$. We derived the Menon-type identity $\sum\limits_{\substack{m=1\\(m.n^s)_s=1}}^{n^s} (m-1,n^s)_s=\Phi_s(n^s)\tau_s(n^s)$ in Czechoslovak Math. J., 72(1):165-176 (2022) where $\Phi_s$ denotes the Klee's function and $(a,b)_s$ denotes a a generalization of the gcd function. Here we give an alternate method to derive this identity using the concept of Cohen-Ramanujan sum.