The Goldston-Pintz-Yildirim Sieve and Maximal Gaps

TL;DR

This paper discusses advances in understanding prime gaps, generalizes key results by Goldston, Pintz, and Yildirim, and improves bounds on maximal gaps in small prime clusters.

Contribution

It generalizes the Goldston-Pintz-Yildirim sieve to broader applications and improves bounds on maximal prime gaps in triplets.

Findings

Generalized prime gap results for broader applications

Improved bounds for maximal gaps in triplets of primes

Enhanced understanding of prime distribution patterns

Abstract

One field of particular interest in Number Theory concerns the gaps between consecutive primes. Within the last few years, very important results have been achieved on how small these gaps can be. The strongest of these results were obtained by Dan Goldston, Janos Pintz and Cem Yalcin Yildirim. The present work begins by generalizing their results so that they can be applied to related problems in a more direct manner. Additionally, we improve the bound for $F_2$ (concerning the maximal gap in a block of three primes) obtained by the authors' first paper with our generalization.