A generalization of Menon's identity with Dirichlet characters

The classical Menon’s identity [7] states that \begin{equation*}\label{oldbegin1} \sum_{\substack{a\in\Bbb Z_n^\ast }}\gcd(a -1,n)=\varphi(n) \sigma_{0} (n), \end{equation*} where for a positive integer $n$, ${\mathbb{Z}}_n^{\ast }$ is the group of units of the ring $\Bbb Z_n=\Bbb Z/n\Bbb Z$,$\gcd(\ ,\ )$representsthegreatestcommondivisor,$\varphi(n)$ is the Euler’s totient function and ${\sigma }_k(n)={\sum }_{d\mathrm{\mid }n}{d}^k$ 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)\chi(a)=\varphi(n)\sigma_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.