Another generalization of Menon’s identity in the ring of algebraic integers
Volume 160 / 2020
Colloquium Mathematicum 160 (2020), 213-221
MSC: Primary 11R04; Secondary 11A25.
DOI: 10.4064/cm7785-6-2019
Published online: 27 January 2020
Abstract
Let $\varphi (n)$ be Euler’s totient function and $\tau (n)$ the divisor function. Recently, Zhao and Cao proved that $$ \sum _{\substack {a=1\atop \gcd (a, n)=1}}^{n} \gcd (a-1, n)\chi (a)=\varphi (n)\tau (n/d), $$ where $\chi $ is a Dirichlet character modulo $n$ with conductor $d$. We generalize the above identity to the ring of algebraic integers by considering arithmetical functions and characters.
