A note on the distribution of normalized prime gaps

Abstract

Let us denote the nth difference between consecutive primes by d_n. The Prime Number Theorem clearly implies that d_n is logn on average. Paul Erd\H{o}s conjectured about 60 years ago that the sequence d_n/logn is everywhere dense on the nonnegative part of the real line. He and independently G. Ricci proved in 1954-55 that the set J of limit points of the sequence {d_n/logn} has positive Lebesgue measure. The first and until now only concrete known element of J was proved to be the number zero in the work of Goldston, Yildirim and the present author. The author of the present note showed in 2013 (arXiv: 1305.6289) that there is a fixed interval containing 0 such that all elements of it are limit points. In 2014 it was shown by W.D. Banks, T. Freiberg and J. Maynard (arXiv: 1404.5094) that one can combine the Erd\H{o}s-Rankin method (producing large prime gaps) and the Maynard-Tao method (producing bounded prime gaps) to obtain the lower bound (1+o(1))T/8 for the Lebesgue measure of the subset of limit points not exceeding T. In the present note we improve this lower bound to (1+o(1))T/4, using a refinement of the argument of Banks, Freiberg and Maynard.