The Ramanujan Journal ( IF 0.7 ) Pub Date : 2021-06-24 , DOI: 10.1007/s11139-021-00448-1 Man Chen , Zhiyong Zheng
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 \({\mathfrak {d}}_i\).
中文翻译：
剩余有限戴德金域中具有狄利克雷特征的梅农型恒等式
本文研究了一般情况下的Menon-Sury 恒等式，即Menon-Sury 恒等式在剩余有限Dedekind 域中涉及Dirichlet 字符。通过使用环的过滤\({\mathfrak {D}}/{\mathfrak {n}}\)及其单位群\(U({\mathfrak {D}}/{\mathfrak {n}}) \)，我们明确计算以下两个总和：
$$\begin{aligned} \sum _{\begin{array}{c} a\in U({\mathfrak {D}}/{\mathfrak {n}}) \\ b_1, \ldots , b_r\in {\mathfrak {D}}/{\mathfrak {n}} \end{array}} N(\langle a-1,b_1, b_2, \ldots, b_r \rangle +{\mathfrak {n}})\chi (a) \end{对齐}$$和
$$\begin{aligned} \sum _{\begin{array}{c} a_{1},\ldots , a_{s}\in U({\mathfrak {D}}/{\mathfrak {n}} ) \\ b_1, \ldots , b_r\in {\mathfrak {D}}/{\mathfrak {n}} \end{array}} N(\langle a_{1}-1,\ldots , a_{s} -1,b_1, b_2, \ldots , b_r \rangle +{\mathfrak {n}})\chi _{1}(a_1) \cdots \chi _{s}(a_s), \end{aligned}$$其中\({\mathfrak {D}}\)是残差有限戴德金域，\({\mathfrak {n}}\)是\({\mathfrak {D}}\)的非零理想，\(N ({\mathfrak {n}})\)是商环的基数\({\mathfrak {D}}/{\mathfrak {n}}\) , \(\chi _{i}~(1\le i\le s)\)是狄利克雷字符 mod \({\mathfrak {n}}\)与导体\({\mathfrak {d}}_i\)。
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