Menon-type identities concerning additive characters (Q2211916)

The classical Menon's identity states that \[ \sum_{\substack{a=1\\ (a,n)=1}}^n \gcd(a-1, n)=\varphi(n)\tau(n), \] where \(\varphi\) is Euler's totient function and \(\tau\) is the divisor number function, see \textit{P.~Kesava Menon} [J. Indian Math. Soc., New Ser. 29, 155--163 (1965; Zbl 0144.27706)]. This identity has been generalized in various directions. \textit{Y. Li} and \textit{D. Kim} [J. Number Theory, 175, 42--50 (2017; Zbl 1407.11009)] present identities for the sum \[ S^*(n,k)=\sum_{\substack{a=1\\ (a,n)=1}}^n (a-1,n)\exp(ka2\pi i/n). \] The present author generalizes these identities considering the sum \[ S_{f}(n,k,s)=\sum_{\substack{a=1\\ (a,n)=1}}^n f_n(a-s)\exp(ka2\pi i/n), \] where \(f_n\) is an even function modulo \(n\), with a different approach, based on convolutional identities. Some other applications, including related formulas for Ramanujan sums, are also discussed.