A Menon-type Identity concerning Dirichlet characters and a generalization of the gcd function

TL;DR

This paper generalizes Menon's identity involving gcd sums and Dirichlet characters by replacing gcd with a broader function, improving previous methods and expanding the identity's applicability.

Contribution

It introduces a new identity replacing gcd with a generalized function, extending Menon's identity and refining existing proof techniques.

Findings

Derived a generalized Menon-type identity with a new function

Improved proof methods for Menon-type identities

Extended the applicability of gcd-related identities

Abstract

Menon's identity is a classical identity involving gcd sums and the Euler totient function $\phi$. In a recent paper, Zhao and Cao derived the Menon-type identity $\sum\limits_{\substack{k=1}}^{n}(k-1,n)\chi(k) = \phi(n)\tau(\frac{n}{d})$, where $\chi$ is a Dirichlet character mod $n$ with conductor $d$. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao-Cao identity can be improved if one uses the method we employ here.