Abstract

Since the Gauss sums are not multiplicative, the Dirichlet series of $n$th order Gauss sums do not have an Euler product. Therefore, they are not expected to satisfy the Riemann Hypothesis. Over $\mathbb{F}_q(t)$, the Dirichlet series of cubic Gauss sums are polynomials in $u = q^{-s}$, which reduces the task of finding the zeros of the series to computing the roots of a polynomial. In this work, we will discuss the challenges that arise when computing these roots and present numerical data on them.