Menon-type identities concerning Dirichlet characters

Abstract

Let $\chi$ be a Dirichlet character (mod $n$) with conductor $d$. In a quite recent paper Zhao and Cao deduced the identity ${\sum }_{k=1}^n(k-1,n)\chi (k)=\phi (n)\tau (n/d)$, 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.

8 pages

References in corpus (4)

• Menon's identity and arithmetical sums representing functions of several variables
• The discrete Fourier transform of $r$-even functions
• A generalization of Menon's identity
• Another generalization of the gcd-sum function