Bias in cubic Gauss sums: Patterson's conjecture

Abstract

Let $W$ be a smooth test function with compact support in $(0,\infty)$. Conditional on the Generalized Riemann Hypothesis for Hecke $L$-functions over $\mathbb{Q}(\omega)$, we prove that $$\sum_{p \equiv 1 \pmod{3}} \frac{1}{2 \sqrt{p}} \cdot \Big ( \sum_{x \pmod{p}} e^{2\pi i x^3 / p} \Big ) W \Big ( \frac{p}{X} \Big ) \sim \frac{(2\pi)^{2/3}}{3 \Gamma(\tfrac 23)} \int_{0}^{\infty} W(x) x^{-1/6} dx \cdot \frac{X^{5/6}}{\log X},$$ as $X \rightarrow \infty$ and $p$ runs over primes. This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846 and confirms (conditionally on the Generalized Riemann Hypothesis) a conjecture of Patterson from 1978. There are two important byproducts of our proof. The first is an explicit level aspect Voronoi summation formula for cubic Gauss sums, extending computations of Patterson and Yoshimoto. Secondly, we show that Heath-Brown's cubic large sieve is sharp up to factors of $X^{o(1)}$ under the Generalized Riemann Hypothesis. This disproves the popular belief that the cubic large sieve can be improved. An important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic main term. This estimate relies on the Generalized Riemann Hypothesis, and is one of the fundamental reasons why our result is conditional.