Bounded intervals containing many primes

TL;DR

This paper combines sieve techniques and recent advances to demonstrate that the minimal gap between primes in certain intervals grows at most exponentially with respect to the interval size.

Contribution

It introduces a novel combination of Harman's sieve method with Maynard and Tao's work to bound prime gaps within bounded intervals.

Findings

Proves a bound on the lim inf of prime gaps within intervals.

Shows the prime gap growth is at most exponential in the interval length.

Advances understanding of prime distribution in bounded regions.

Abstract

By combining a sieve method of Harman with the work of Maynard and Tao we show that $$\liminf_{n\rightarrow \infty}(p_{n+m}-p_n)\ll \exp(3.815m).$$