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

Big biases amongst products of two primes

David Dummit and Hershy Kisilevsky observed from calculation that the Legendre symbols (p/q) and (q/p) are unequal for rather more than a quarter of the pairs of odd primes p and q with pq<=x, during some calculations. In fact almost 30% of the pq's up to a million satisfy p=q=3 mod 4. Together we found that this is no accident and that the bias up to x is roughly 1 +1/3(loglog x-1). This is a much stronger bias than the traditional "prime race" problem. When doing the math one finds that this problems about pq's is equivalent to the prime race problem, for primes =3 mod 4 versus those =1 mod 4, in which we weight each prime by its reciprocal.

On ranks of quadratic twists of elliptic curve

Gaps between fractional parts, and additive combinatorics

The frequency and the structure of large character sums

When the sieve works

An accurate running time estimate for the quadratic sieve

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

    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?