Menon-type identities with additive characters

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.