Menon-type identities with additive characters

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([Formula presented]), 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⁡([Formula presented])=exp⁡([Formula presented])τ(gcd⁡(k,n))φ(n) if ord_p(n)−ord_p(k)≠1 holds for any prime p dividing n, where for u∈Z, ord_p(u) is the exponent of the highest power of p dividing u. For k=0, our result reduces to the classical Menon's identity.