A Menon-type identity in residually finite Dedekind domains (Q266583)

Menon's identity states that NEWLINE\[NEWLINE \sum_{k=1\atop (k, n)=1}^{n} (k-1, n)=\varphi(n)\tau(n), NEWLINE\]NEWLINE where \(\varphi\) is Euler's totient function and \(\tau\) is the divisor number function, see \textit{P. Kesava Menon} [J. Indian Math. Soc., New Ser. 29, 155--163 (1965; Zbl 0144.27706)]. This identity has been generalized by many authors, see e.g. \textit{B. Sury} [Rend. Circ. Mat. Palermo (2) 58, No. 1, 99--108 (2009; Zbl 1187.20015)]. For a survey of generalizations see \textit{L. Tóth} [Rend. Semin. Mat., Univ. Politec. Torino 69, No. 1, 97--110 (2011; Zbl 1235.11011)].NEWLINENEWLINEIn [J. Number Theory 137, 179--185 (2014; Zbl 1293.11008)], the present author extends Menon's identity to residually finite Dedekind domains, that is, to Dedekind domains \(\mathfrak{D}\) such that for each non-zero ideal \(\mathfrak{n}\) of \(\mathfrak{D}\), the residue class ring \(\mathfrak{D}/\mathfrak{n}\) is finite. In the paper under review the present author extends Sury's identity to residually finite Dedekind domains.