Menon-type identities with additive characters

Abstract

The classical Menon's identity [7] states that∑a=1gcd⁡(a,n)=1ngcd⁡(a−1,n)=φ(n)τ(n),for every positive integer n, where φ(n) is the Euler's totient function and τ(n) is the number of positive divisors of n. Recently, Zhao and Cao [19] extended Menon's identity to Dirichlet characters:∑a=1gcd⁡(a,n)=1ngcd⁡(a−1,n)χ(a)=φ(n)τ(nd),where χ is a Dirichlet character modulo n and d is the conductor of χ. In this paper, we extend Menon's identity to additive characters. A special case of our main result reads like this:∑a=1gcd⁡(a,n)=1ngcd⁡(a−1,n)exp⁡(ka2πin)=exp⁡(k2πin)τ(gcd⁡(k,n))φ(n) if ordp(n)−ordp(k)≠1 holds for any prime p dividing n, where for u∈Z, ordp(u) is the exponent of the highest power of p dividing u. For k=0, our result reduces to the classical Menon's identity.