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

Menon’s identity is a classical identity involving gcd sums and the Euler totient function ϕ . In a recent paper, Zhao and Cao ( Int. J. Number Theory 13 ( 9 ) (2017) 2373–2379) derived the Menon-type identity ∑ k = 1 n ( k - 1 , n ) χ ( k ) = ϕ ( n ) τ ( n d ) , 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.