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Pretentiousness in analytic number theory

Andrew Granville — 2009

Journal de Théorie des Nombres de Bordeaux

In this report, prepared specially for the program of the , we describe how, in joint work with K. Soundararajan and Antal Balog, we have developed the notion of “pretentiousness” to help us better understand several key questions in analytic number theory.

Solution to a problem of Bombieri

Andrew Granville — 1993

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We solve a problem of Bombieri, stated in connection with the «prime number theorem» for function fields.

Zeros of Fekete polynomials

Brian ConreyAndrew GranvilleBjorn PoonenK. Soundararajan — 2000

Annales de l'institut Fourier

For p an odd prime, we show that the Fekete polynomial f p ( t ) = a = 0 p - 1 a p t a has κ 0 p zeros on the unit circle, where 0 . 500813 > κ 0 > 0 . 500668 . Here κ 0 - 1 / 2 is the probability that the function 1 / x + 1 / ( 1 - x ) + n : n 0 , 1 δ n / ( x - n ) has a zero in ] 0 , 1 [ , where each δ n is ± 1 with y 1 / 2 . In fact f p ( t ) has absolute value p at each primitive p th root of unity, and we show that if | f p ( e ( 2 i π ( K + τ ) / p ) ) | < ϵ p for some τ ] 0 , 1 [ then there is a zero of f close to this arc.

Ranks of quadratic twists of elliptic curves

Mark WatkinsStephen DonnellyNoam D. ElkiesTom FisherAndrew GranvilleNicholas F. Rogers — 2014

Publications mathématiques de Besançon

We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find...

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