Large gaps between consecutive prime numbers

Abstract

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X)\ge f(X)\frac{\log X\log \log X\log \log \log \log X}{(\log \log \log X{)}^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.

v2. very minor corrections. To appear in Ann. Math

References in corpus (2)

• Gowers norms, regularization and limits of functions on abelian groups
• Long gaps between primes

Cited by in corpus (5)

• Primes in intervals of bounded length
• Polynomial patterns in the primes
• Geometric Progression-Free Sequences with Small Gaps
• A higher-dimensional Siegel-Walfisz theorem
• Note On The Maximal Primes Gaps