A generalization of Menon's identity with Dirichlet characters

The classical Menon's identity (P. K. Menon, On the sum Sigma(a - 1, n) [(a, n) = 1] J. Indian Math. Soc. (N.S.) 29 (1965) 155-163] states that Sigma(a is an element of Zn*) gcd (a-1, n) = phi(n)sigma(0)(n), where for a positive integer n, Z(n)* is the group of units of the ring Z(n) = Z/nZ, gcd(, ) represents the greatest common divisor, phi(n) is the Euler's totient function and sigma(k)(n) = Sigma(d vertical bar n) d(k) is the divisor function. In this paper, we generalize Menon's identity with Dirichlet characters in the following way: [GRAPHICS] gcd (a - 1, b(1), ..., b(k), n)chi(a) = phi(n)sigma(k)(n/d), where k is a non-negative integer and chi 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.