A generalization of Menon’s identity with Dirichlet characters

The classical Menon’s identity [P. K. Menon, On the sum [Formula: see text], J. Indian Math. Soc.[Formula: see text]N.S.[Formula: see text] 29 (1965) 155–163] states that [Formula: see text] where for a positive integer [Formula: see text], [Formula: see text] is the group of units of the ring [Formula: see text], [Formula: see text] represents the greatest common divisor, [Formula: see text] is the Euler’s totient function and [Formula: see text] is the divisor function. In this paper, we generalize Menon’s identity with Dirichlet characters in the following way: [Formula: see text] where [Formula: see text] is a non-negative integer and [Formula: see text] is a Dirichlet character modulo [Formula: see text] whose conductor is [Formula: see text]. Our result can be viewed as an extension of Zhao and Cao’s result [Another generalization of Menon’s identity, Int. J. Number Theory 13(9) (2017) 2373–2379] to [Formula: see text]. It can also be viewed as an extension of Sury’s result [Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo 58 (2009) 99–108] to Dirichlet characters.