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Multiple Gauss sums

Jianya Liu, Sizhe Xie

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 $s \geq D^2 4^{D+2} R^5$.

Multiple Gauss sums

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 be forms with differing degrees, with being the highest degree, and let be nonsingular. We prove that the system is solvable in primes provided that .

Paper Structure

This paper contains 7 sections, 7 theorems, 65 equations.

Table of Contents

  1. Introduction and statement of results
  2. Known results
  3. Complete exponential sums
  4. Main result
  5. Application to the Birch--Goldbach problem
  6. Geometric considerations
  7. Proof of Theorem \ref{['prop gen Gau']}

Key Result

Proposition 1.1

Let $\xi$ be a complex primitive $q$-th root of unity, $F\in \mathbb{Z}[x_1,\ldots,x_s]$ with critical locus of dimension $\Delta$, and define the exponential sum Then there exists a positive constant $\Theta=\Theta_d$ depending on the degree $d$ of $F$ such that where $\varepsilon>0$ is arbitrarily small. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (12)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 2 more