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 involving Dirichlet characters in residually finite Dedekind domains. By using the filtration of the ring \({\mathfrak {D}}/{\mathfrak {n}}\) and its unit group \(U({\mathfrak {D}}/{\mathfrak {n}})\), we explicitly compute the following two summations:

$$\begin{aligned} \sum _{\begin{array}{c} a\in U({\mathfrak {D}}/{\mathfrak {n}}) \\ b_1, \ldots , b_r\in {\mathfrak {D}}/{\mathfrak {n}} \end{array}} N(\langle a-1,b_1, b_2, \ldots , b_r \rangle +{\mathfrak {n}})\chi (a) \end{aligned}$$

and

$$\begin{aligned} \sum _{\begin{array}{c} a_{1},\ldots , a_{s}\in U({\mathfrak {D}}/{\mathfrak {n}}) \\ b_1, \ldots , b_r\in {\mathfrak {D}}/{\mathfrak {n}} \end{array}} N(\langle a_{1}-1,\ldots , a_{s}-1,b_1, b_2, \ldots , b_r \rangle +{\mathfrak {n}})\chi _{1}(a_1) \cdots \chi _{s}(a_s), \end{aligned}$$

where \({\mathfrak {D}}\) is a residually finite Dedekind domain and \({\mathfrak {n}}\) is a nonzero ideal of \({\mathfrak {D}}\), \(N({\mathfrak {n}})\) is the cardinality of quotient ring \({\mathfrak {D}}/{\mathfrak {n}}\), \(\chi _{i}~(1\le i\le s)\) are Dirichlet characters mod \({\mathfrak {n}}\) with conductor \({\mathfrak {d}}_i\).