Andrew Granville's Home Page


Montreal Number Theory webpage (including Quebec-Vermont seminar, and analytic number theory seminar)

Number Theory, special year at the CRM, 2014-15: From Arithmetic Statistics to Zeta elements

Other Seminars


Course Notes

Useful references

Number theory journals

My Recent Research

On ranks of quadratic twists of elliptic curve

Some years ago I presented a heuristic that, in the family of quadratic twists of a given elliptic curve, the rank is absolutely bounded, the proposed bound depending only on the number of rational 2-torsion points. At the time this contradicted the popular belief. Mark Watkins took it upon himself to do a massive calculation of ranks of quadratic twists of the congruent number curve, to test out my "conjecture". This paper is the record of an enormous calculation, performed under Mark's leadership, involving the incredibly sophisticated algorithms and ideas of the other co-authors (Stephen Donnelly, Noam Elkies, Tom Fisher,and Nick Rogers). The evidence is as compelling as we have any right to hope for, suggesting that the quadratic twists all have rank less than or equal to 7.

When the sieve works

The number of sumsets in a finite field

An accurate running time estimate for the quadratic sieve

The central limit theorem and the Chinese Remainder Theorem

Selected expository articles

  • Primes in Intervals of Bounded Length
  • Basics of binary quadratic forms and Gauss composition
  • Don't be seduced by the zeros!
  • Different approaches to the distribution of primes
  • The Princeton Companion to Mathematics: Analytic number theory
  • Prime number patterns (2009 Ford Prize)
  • It is easy to determine whether a given integer is prime (2008 Chauvenet Prize)
  • Prime number races (2007 Ford Prize)
  • It's as easy as abc
  • Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle (1995 Hasse Prize)
  • Publications

    Short resumé

    Upcoming Travel Calendar

    Sept 15-19, 2014: Statistics and number theory, CRM, Montreal
    Sept 22-26, 2014: LMS-CMI Research school on Bounded Gaps Between Primes, Oxford
    Sept 29-Oct 3, 2014: CMI workshop, Analytic number theory, Oxford
    Oct 6-10, 2104: Additive Combinatorics and expanders, CRM, Montreal
    Nov 10-14, 2014: Counting arithmetic objects, CRM, Montreal
    Dec 8-12, 2014: Probabilistic and multiplicative number theory, CRM Montreal

    Contact Information

    Need a reference?

    Public lectures

    Primes, including an animation

    MSI: Anatomy of Integers and Permutations

    Of general interest

    Many of the papers of Paul Erdos

    CRM-ISM Postdoctoral Fellowships

    Best Professions in 2010?

    Thanking our sponsors

    Differences in pure and applied math and stats?