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Bias in cubic Gauss sums: Patterson’s conjecture (Alex Dunn, CalTech)
April 25, 2023 @ 12:15 pm - 1:10 pm
We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann Hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias in the distribution of cubic Gauss sums first 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.