Bounded gaps between primes in short intervals

Abstract

Abstract Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form

$$[x-x^{0.525},x]$$

[ 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

$$\delta \in [0.525,1]$$

δ ∈ [ 0.525 , 1 ]

there exist positive integers k, d such that for sufficiently large x, the interval

$$[x-x^\delta ,x]$$

[ x -

x δ

, x ]

contains

$$\gg _{k} \frac{x^\delta }{(\log x)^k}$$

≫ k

x δ

( 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.