Gauss sum with principal multiplicative character

Abstract

Let $R$ be a finite ring with unity, $\ddot{I}ˆ:R\to {\mathbb{C}}^{\times }$ be an additive character of $R$, and \( χ_0 \) be the principal multiplicative character ($i.e.$, $\ddot{I}{‡}_0(x)=1\text{for all }x\in R^{\times }$), then the Gauss sum is \[ G(χ_0, ψ) = \sum_{x \in R^\times} ψ(x). \] In this paper, we give an explicit formula for a more general form of the Gauss sum $G(\ddot{I}{‡}_0,\ddot{I}ˆ)$. Interestingly, the formula extends the known formula of classical Ramanujan's sum to the context of finite rings. As an application, we derive the eigenvalues for a more general form of the unitary Cayley graph $\text{Cay}(R,R^{\times })$ using the formula.