A Menon-type Identity with Multiplicative and Additive Characters

This paper studies Menon-type identities involving both multiplicative characters and additive characters. In the paper, we shall give the explicit formula of the following sum Sigma gcd(a - 1, b(1), ..., b(k), n) chi(a)lambda(1)(b(1)) ... lambda(k) (b(k)), a is an element of Z(n)* b(1), ..., b(k)is an element of Z(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, chi is a Dirichlet character modulo n, and for nonnegative integer k, lambda(1),..., lambda(k) are additive characters of Z(n). Our formula further extends the previous results by Sury [13], Zhao-Cao [17] and Li-Hu-Kim [4].