BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alex Dunn (Caltech) DTSTART:20220207T230000Z DTEND:20220207T235000Z DTSTAMP:20260423T041611Z UID:UCLA_NTS/45 DESCRIPTION:Title: Bias in cubic Gauss sums: Patterson's conjecture\nby Alex Dunn (Calt ech) as part of UCLA Number Theory Seminar\n\nLecture held in Math Science 5118.\n\nAbstract\nWe prove\, in this joint work with Maksym Radziwill\, a 1978 conjecture of S. Patterson (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 first observe d by Kummer in 1846.\n\nThere are two important byproducts of our proof. T he first is an explicit level aspect Voronoi summation formula for cubic G auss sums\, extending computations of Patterson and Yoshimoto. Secondly\, we show that Heath-Brown's cubic large sieve is sharp under GRH. This disp roves the popular belief that the cubic large sieve can be improved.\n\nAn important ingredient in our proof is a dispersion estimate for cubic Gaus s sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic main term. This estimate relies on the Generalised Riemann Hypothesis\, and is one of the fundamental reasons why our result is conditional.\n LOCATION:https://researchseminars.org/talk/UCLA_NTS/45/ END:VEVENT END:VCALENDAR