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

Abstract

Menon's identity is a classical identity involving gcd sums and the Euler totient function $\varphi$. In a recent paper, Zhao and Cao derived the Menon-type identity $\sum _{\begin{array}{c}k=1\end{array}}^n(k-1,n)\chi (k)=\varphi (n)\tau (\frac{n}{d})$, 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.