Multiple Gauss sums

Jianya Liu; Sizhe Xie

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arXiv · 2604.03347

Multiple Gauss sums

Jianya Liu·Sizhe Xie

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Abstract

A multiple Gauss sum is a complete multiple exponential sum twisted by Dirichlet characters. We prove a new bound for multiple Gauss sums and, as an application, improve previous results in the Birch--Goldbach problem. Let $F_1, \ldots, F_R \in \mathbb{Z}[x_1, \ldots, x_s]$ be forms with differing degrees, with $D$ being the highest degree, and let $\boldsymbol{F} = (F_1, \ldots, F_R)$ be nonsingular. We prove that the system $\boldsymbol{F}(\boldsymbol{x})=\mathbf{0}$ is solvable in primes provided that $s \geq D^2 4^{D+2} R^5$.

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Jianya Liu, Sizhe Xie. 2026-04-03. Multiple Gauss sums. https://arxiv.org/abs/2604.03347

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