A generalization of Menon's identity with Dirichlet characters

Abstract

The classical Menon's identity [7] states that \begin{equation*}\label{oldbegin1} \sum_{\substack{a\in\Bbb Z_n^\ast }}\gcd(a -1,n)=φ(n) σ_{0} (n), \end{equation*} where for a positive integer $n$, ${\mathbb{Z}}_n^{\ast }$ is the group of units of the ring ${\mathbb{Z}}_n=\mathbb{Z}/n\mathbb{Z}$, $\gcd (,)$ represents the greatest common divisor, $\phi (n)$ is the Euler's totient function and ${\sigma }_k(n)={\sum }_{d|n}{d}^k$ is the divisor function. In this paper, we generalize Menon's identity with Dirichlet characters in the following way: \begin{equation*} \sum_{\substack{a\in\Bbb Z_n^\ast b_1, ..., b_k\in\Bbb Z_n}} \gcd(a-1,b_1, ..., b_k, n)χ(a)=φ(n)σ_k\left(\frac{n}{d}\right), \end{equation*} where $k$ is a non-negative integer and $\chi$ is a Dirichlet character modulo $n$ whose conductor is $d$. Our result can be viewed as an extension of Zhao and Cao's result [16] to $k>0$. It can also be viewed as an extension of Sury's result [12] to Dirichlet characters.

8 pages