**
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