EUDML

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.

Limitations to the equi-distribution of primes III

John Friedlander; Andrew Granville — 1992

Compositio Mathematica

On Sophie Germain type criteria for Fermat's Last Theorem

Andrew Granville; Barry Powell — 1988

Acta Arithmetica

The number of unsieved integers up to x

Andrew Granville; K. Soundararajan — 2004

Acta Arithmetica

Rabinowitsch revisited

Andrew Granville; Richard A. Mollin — 2000

Acta Arithmetica

The square of the Fermat quotient.

Granville, Andrew — 2004

Integers

Cycle lengths in a permutation are typically Poisson.

Granville, Andrew — 2006

The Electronic Journal of Combinatorics [electronic only]

Zeros of Fekete polynomials

Brian Conrey; Andrew Granville; Bjorn Poonen; K. Soundararajan — 2000

Annales de l'institut Fourier

For $p$ an odd prime, we show that the Fekete polynomial $f_{p} (t) = \sum_{a = 0}^{p - 1} (\frac{a}{p}) t^{a}$ has $\sim κ_{0} p$ zeros on the unit circle, where $0.500813 > κ_{0} > 0.500668$ . Here $κ_{0} - 1 / 2$ is the probability that the function $1 / x + 1 / (1 - x) + \sum_{n \in ℤ : n \neq 0, 1} δ_{n} / (x - n)$ has a zero in $] 0, 1 [$ , where each $δ_{n}$ is $\pm 1$ with y $1 / 2$ . In fact $f_{p} (t)$ has absolute value $\sqrt{p}$ at each primitive $p$ th root of unity, and we show that if $| f_{p} (e (2 i π (K + τ) / p)) | < ϵ \sqrt{p}$ for some $τ \in] 0, 1 [$ then there is a zero of $f$ close to this arc.

Ranks of quadratic twists of elliptic curves

Mark Watkins; Stephen Donnelly; Noam D. Elkies; Tom Fisher; Andrew Granville; Nicholas 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...