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 \(\phi \). In a recent paper, Zhao and Cao (Int. J. Number Theory 13(9) (2017) 2373–2379) derived the Menon-type identity \(\sum _{\begin{array}{c} k=1 \end{array}}^{n}(k-1,n)\chi (k) = \phi (n)\tau (\frac{n}{d})\), 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.