A note on the distribution of normalized prime gaps

TL;DR

This paper improves the lower bound on the measure of limit points of normalized prime gaps, combining advanced methods to show that a significant portion of these points are densely distributed.

Contribution

It refines previous bounds on the measure of limit points of normalized prime gaps by enhancing the combination of existing methods.

Findings

Lower bound on measure of limit points improved to (1+o(1))T/4

Demonstrates dense distribution of normalized prime gaps

Builds on and refines methods by Banks, Freiberg, and Maynard

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.