A generalization of Menon's identity with Dirichlet characters

The classical Menon's identity [P. K. Menon, On the sum Σ(a - 1, n)[(a, n) = 1], J. Indian Math. Soc. (N.S.) 29 (1965) 155-163] states that (Formula Presented) where for a positive integer n, ℤn is the group of units of the ring ℤn = ℤ/nℤ, gcd(, ) represents the greatest common divisor, φ(n) is the Euler's totient function and σk(n) = Σ d|n dk is the divisor function. In this paper, we generalize Menon's identity with Dirichlet characters in the following way: (Formula Presented) where k is a non-negative integer and χ is a Dirichlet character modulo n whose conductor is d. 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 k > 0. 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.