math.NT

(what is this?)

# Title: Bounded gaps between primes in short intervals

Abstract: Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $[x-x^{0.525},x]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any $\delta\in [0.525,1]$ there exist positive integers $k,d$ such that for sufficiently large $x$, the interval $[x-x^\delta,x]$ contains $\gg_{k} \frac{x^\delta}{(\log x)^k}$ pairs of consecutive primes differing by at most $d$. This confirms a speculation of Maynard that results on small gaps between primes can be refined to the setting of short intervals of this length.
 Comments: 29 pages Subjects: Number Theory (math.NT) MSC classes: 11N05, 11N36 Cite as: arXiv:1707.05437 [math.NT] (or arXiv:1707.05437v1 [math.NT] for this version)

## Submission history

From: Sammy Luo [view email]
[v1] Tue, 18 Jul 2017 02:00:47 GMT (22kb)