For more information about this meeting, contact Robert Vaughan.

Title: | The distribution of the variance of primes in arithmetic progressions |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Daniel Fiorilli, University of Michigan |

Abstract: |

Gallagher's refinement of the Barban-Davenport-Halberstam
states that V(x;q), the variance of primes up to x in the arithmetic
progressions modulo q, is at most x log q, on average over q in the
range x/(log x)^A < q < x. It was then discovered by Montgomery that
in this range V(x;q) is actually asymptotic to x log q (on average
over q); his result was refined by a long list of authors including
Hooley, Goldston and Vaughan, and Friedlander and Goldston. Tools used
in these papers include the circle method and divisor switching
techniques, and under GRH and a strong from of the Hardy-Littlewood Conjecture it is now known that V(x;q) is asymptotic to x log q in the range x^{1/2+o(1)} < q< x. While it is not clear that the asymptotic should hold for more moderate values of q, Keating and Rudnick have proven an estimate for the function field analogue of V(x;q) which suggests that this range could be extended to x^{o(1)} < q< x. In this talk we will show how one can use probabilistic techniques to give evidence that V(x;q) should be asymptotic to x log q in the even wider range (log log x)^{1+o(1)} < q < x, and that this range is best possible. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 04 / 11 / 2013 |

Time: | 11:15am - 12:05pm |