Bounded gaps between primes in short intervals
Author(s)
Alweiss, Ryan; Luo, Sammy
Download40993_2018_109_ReferencePDF.pdf (373.4Kb)
Publisher Policy
Publisher Policy
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Terms of use
Metadata
Show full item recordAbstract
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.
Date issued
2018-03-19Department
Massachusetts Institute of Technology. Department of MathematicsPublisher
Springer International Publishing
Citation
Research in Number Theory. 2018 Mar 19;4(2):15
Version: Author's final manuscript