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 φ. In a recent paper, Zhao and Cao derived the Menon-type identity n k=1, where χ 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.