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.
Relational Format
presentation
Recommended Citation
Dunn, Alex, "Bias in cubic Gauss sums: Patterson’s conjecture" (2022). Algebra/Number Theory Seminar. 13.
https://egrove.olemiss.edu/math_algebra_number_theory/13