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.