Bounded gaps between primes in short intervals
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Author(s)
Alweiss, Ryan
Luo, Sammy
Date Issued
March 19, 2018
Publisher
Springer International Publishing
Citation
Research in Number Theory. 2018 Mar 19;4(2):15
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s40993-018-0109-y
