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 $$[x-x^{0.525},x]$$...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer International Publishing
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131478 |
| _version_ | 1826203957330968576 |
|---|---|
| author | Alweiss, Ryan Luo, Sammy |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Alweiss, Ryan Luo, Sammy |
| author_sort | Alweiss, Ryan |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T12:46:07Z |
| format | Article |
| id | mit-1721.1/131478 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:46:07Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1314782023-11-07T20:05:06Z Bounded gaps between primes in short intervals Alweiss, Ryan Luo, Sammy Massachusetts Institute of Technology. Department of Mathematics 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. 2021-09-20T17:17:14Z 2021-09-20T17:17:14Z 2018-03-19 2020-09-24T21:18:24Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131478 Research in Number Theory. 2018 Mar 19;4(2):15 en https://doi.org/10.1007/s40993-018-0109-y 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. SpringerNature application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Alweiss, Ryan Luo, Sammy Bounded gaps between primes in short intervals |
| title | Bounded gaps between primes in short intervals |
| title_full | Bounded gaps between primes in short intervals |
| title_fullStr | Bounded gaps between primes in short intervals |
| title_full_unstemmed | Bounded gaps between primes in short intervals |
| title_short | Bounded gaps between primes in short intervals |
| title_sort | bounded gaps between primes in short intervals |
| url | https://hdl.handle.net/1721.1/131478 |
| work_keys_str_mv | AT alweissryan boundedgapsbetweenprimesinshortintervals AT luosammy boundedgapsbetweenprimesinshortintervals |