Gauss sum with principal multiplicative character

Abstract

Let $R$ be a finite ring with unity, $ψ: R \to \mathbb{C}^\times$ be an additive character of $R$, and \( χ_0 \) be the principal multiplicative character ($i.e.$, $χ_0(x) = 1 \quad \text{for all } x \in R^\times$), 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 $G(χ_0, ψ)$. 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.