# Menon-type identities concerning Dirichlet characters

## TL;DR

This paper generalizes Menon's identity for Dirichlet characters by considering even functions modulo n, providing new formulas and an alternative proof approach, and explores related Ramanujan sum identities.

## Contribution

It extends Menon's identity to even functions modulo n and offers a new proof method, also deriving related formulas involving Ramanujan sums.

## Key findings

- Generalized Menon-type identities for even functions mod n
- Derived new formulas involving Ramanujan sums
- Provided an alternative proof approach for the identities

## Abstract

Let $\chi$ be a Dirichlet character (mod $n$) with conductor $d$. In a quite recent paper Zhao and Cao deduced the identity $\sum_{k=1}^n (k-1,n) \chi(k)= \varphi(n)\tau(n/d)$, which reduces to Menon's identity if $\chi$ is the principal character (mod $n$). We generalize the above identity by considering even functions (mod $n$), and offer an alternative approach to proof. We also obtain certain related formulas concerning Ramanujan sums.