Menon-type identities concerning Dirichlet characters.

Let be a Dirichlet character (mod ) with conductor . In a quite recent paper Zhao and Cao deduced the identity , which reduces to Menon's identity if is the principal character (mod ). We generalize the above identity by considering even functions (mod ), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums. [ABSTRACT FROM AUTHOR]