PSU Mathematics Department - Meeting Details

For more information about this meeting, contact Robert Vaughan.

Abstract:
Title:	The distribution of the variance of primes in arithmetic progressions
Seminar:	Algebra and Number Theory Seminar
Speaker:	Daniel Fiorilli, University of Michigan
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.