Menon-type identities concerning Dirichlet characters

Abstract: Let $\chi$ be a Dirichlet character (mod $n$) with conductor $d$. In a quite paper Zhao and Cao deduced the identity $\sum_{k=1}^n (k-1,n) \chi(k)= \varphi(n)\tau(n/d)$, which reduces to Menon's identity if $\chi$ is the principal character (mod $n$). We generalize the above identity by considering even functions (mod $n$), and offer an alternative approach to proof. We also obtain certain related formulas concerning Ramanujan sums.