On the frequency of small gaps between the primes

TL;DR

This paper provides an unconditional analysis of the frequency of small gaps between primes, advancing understanding beyond previous conditional results based on deep hypotheses.

Contribution

It offers an unconditional result on the distribution of small prime gaps, weakening the assumptions needed compared to prior work under the Hardy-Littlewood conjecture.

Findings

Unconditional bounds on small prime gaps

Extension of previous conditional results

Insights into the divergence of reciprocal sums of primes

Abstract

In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime $k$-tuple conjecture, he gave an almost precise answer for it. In the present work we give an unconditional answer for a much weaker form of the same problem.