"Bias in cubic Gauss sums: Patterson’s conjecture" by Alex Dunn
 

Bias in cubic Gauss sums: Patterson’s conjecture

Document Type

Lecture

Publication Date

11-2-2022

Abstract

We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patter-son (conditional on the Generalised Riemann hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias in the distribution of cubic Gauss sums rst observed by Kummer in 1846. One important byproduct of our proof is that we show Heath-Brown’s cubic large sieve is sharp under GRH. 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.

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