Menon-type identities concerning Dirichlet characters

Abstract

Let χ be a Dirichlet character (mod n) with conductor d. In a quite recent paper Zhao and Cao deduced the identity n k=1 (k − 1, n)χ(k) = ϕ(n)τ (n/d), which reduces to Menon's identity if χ 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.

Menon-type identities concerning additive characters

“…Our results generalize those by Li and Kim [2]. We use a different approach, similar to our paper [3], based on certain convolutional identities valid for every n ∈ N.…”

Short proof and generalization of a Menon-type identity by Li, Hu and Kim

“…If χ is the principal character (mod n), that is d = 1, then (1.2) reduces to Menon's identity (1.1). Generalizations of (1.2) involving even functions (mod n) were deduced by the author [6], using a different approach.…”

Proofs, generalizations and analogs of Menon's identity: a survey

“…Generalizations of (4.26) involving n-even functions have been deduced by Tóth [84], using a different approach based on direct manipulations of the corresponding sums, valid for any integer n ∈ N. We quote here only the following result, having a simple proof. See [84,Th. 2.7].…”