A Menon-type identity with Dirichlet characters in residually finite Dedekind domains

This paper studies Menon-Sury's identity in a general case, i.e., the Menon-Sury's identity involvingDirichlet characters in residually finite Dedekind domains. By using the filtration of the ring D/n and its unit group U(D/n), we explicitly compute the following two summations: Sigma(a is an element of U(D/n)) b(1),...,b(r) is an element of D/n N(< a - 1, b(1), b(2), ... , b(r)> + n)chi(a) and Sigma a(1),...,a(s) is an element of U(D/n) b(1), ... ,b(r) is an element of D/ N(< a(1) - 1,..., a(s) - 1, b(1), b(2), ... , b(r)> + n)chi(1)(a(1)) center dot center dot center dot chi(s) (a(s)), where D is a residually finite Dedekind domain and n is a nonzero ideal of D, N(n) is the cardinality of quotient ring D/n, chi(i) (1 <= i <= s) are Dirichlet characters mod n with conductor di