Primes in Tuples IV: Density of small gaps between consecutive primes

TL;DR

This paper demonstrates that a positive proportion of gaps between consecutive primes are small, providing both unconditional and conditional quantitative results on the density of such small gaps.

Contribution

It establishes that small prime gaps occur with positive density and offers new quantitative insights, some proven unconditionally and others conditionally.

Findings

A positive proportion of prime gaps are small.

Quantitative bounds on the density of small gaps.

Results include both unconditional and conditional cases.

Abstract

We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.