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:

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

and

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

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}~(1le ile s)) are Dirichlet characters mod ({mathfrak {n}}) with conductor ({mathfrak {d}}_i).