Large gaps between primes

TL;DR

This paper demonstrates the existence of infinitely many pairs of consecutive primes with gaps larger than a specified function involving multiple iterated logarithms, advancing understanding of prime distribution.

Contribution

It introduces a novel approach combining recent sieve method progress with the Erdos-Rankin construction to establish large prime gaps, answering a longstanding question of Erdos.

Findings

Existence of large prime gaps exceeding a complex logarithmic function

Integration of modern sieve techniques with classical constructions

Progress towards understanding the distribution of prime gaps

Abstract

We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.