## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, October 22, 2015 — 1:30 PM EDT

**Tristan Freiberg, Department of Pure Mathematics, University of Waterloo**

“The distribution of primes in short intervals.”

Cram ́er’s random model leads us to expect that the primes are distributed in a Poisson distribution around their mean spacing. For instance, it is conjectured that for any given positive real number λ and nonnegative integer m, the proportion of positive integers n ≤ x for which the interval (n, n + λ log n] contains exactly m primes is asymptotically equal to λme−λ/m! as x tends to infinity. We show that the number of such n is at least x1−o(1). The proof combines the breakthrough work of Maynard and Tao on short gaps between primes with an Erd ̋os–Rankin type construction for producing large gaps between consecutive primes. This is a continuation of earlier work with Banks and Maynard on limit points of normalized “level spacings” in the sequence of primes, to which most of our discussion will be devoted.

MC 5479

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1