Gauss sum with principal multiplicative character

TL;DR

This paper derives an explicit formula for Gauss sums with principal multiplicative characters over finite rings, extending Ramanujan's sum and applying it to eigenvalues of generalized Cayley graphs.

Contribution

It provides a new explicit formula for Gauss sums over finite rings with principal characters, extending classical results to a broader algebraic context.

Findings

Explicit formula for Gauss sum over finite rings with principal characters

Extension of Ramanujan's sum to finite ring context

Eigenvalues of generalized unitary Cayley graphs derived from the formula

Abstract

Let $R$ be a finite ring with unity, $\psi: R \to \mathbb{C}^\times$ be an additive character of $R$, and \( \chi_0 \) be the principal multiplicative character ($i.e.$, $\chi_0(x) = 1 \quad \text{for all } x \in R^\times$), then the Gauss sum is \[ G(\chi_0, \psi) = \sum_{x \in R^\times} \psi(x). \] In this paper, we give an explicit formula for a more general form of the Gauss sum $G(\chi_0, \psi)$. 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.