Journal article
Almost primes in almost all short intervals
- Abstract:
- Let Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵ x] contain E 3 numbers, and almost all intervals [x,x + log3.51 x] contain E 2 numbers. By this we mean that there are only o(X) integers 1 ⩽ x ⩽ X for which the mentioned intervals do not contain such numbers. The result for E 3 numbers is optimal up to the ϵ in the exponent. The theorem on E 2 numbers improves a result of Harman, which had the exponent 7 + ϵ in place of 3.51. We also consider general Ek numbers, and find them on intervals whose lengths approach log x as k → ∞.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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- Files:
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(Preview, Accepted manuscript, pdf, 631.4KB, Terms of use)
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- Publisher copy:
- 10.1017/S0305004116000232
Authors
- Publisher:
- Cambridge University Press
- Journal:
- Mathematical Proceedings More from this journal
- Volume:
- 161
- Issue:
- 2
- Pages:
- 247-281
- Publication date:
- 2016-04-13
- Acceptance date:
- 2016-03-07
- DOI:
- EISSN:
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1469-8064
- ISSN:
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0305-0041
- Keywords:
- Pubs id:
-
pubs:935446
- UUID:
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uuid:c5be1717-1c47-4927-a96a-8f823d35a2be
- Local pid:
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pubs:935446
- Source identifiers:
-
935446
- Deposit date:
-
2018-10-30
- ARK identifier:
Terms of use
- Copyright holder:
- Cambridge Philosophical Society
- Copyright date:
- 2016
- Notes:
- Copyright: © Cambridge Philosophical Society 2016. This is the accepted manuscript version of the article. The final version is available online from Cambridge University Press at: https://doi.org/10.1017/S0305004116000232
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