Positive Proportion of Small Gaps Between Consecutive Primes

TL;DR

This paper proves that a positive proportion of consecutive prime gaps are smaller than any fixed fraction of the average prime spacing, highlighting the frequent occurrence of small prime gaps.

Contribution

It establishes that a positive proportion of prime gaps are short, extending our understanding of the distribution of small gaps between primes.

Findings

A positive proportion of prime gaps are short.

Short gaps occur with positive density among all prime gaps.

The result holds for any fixed fraction of the average prime gap.

Abstract

We prove that a positive proportion of the gaps between consecutive primes are short gaps of length less than any fixed fraction of the average spacing between primes.