Almost primes in almost all short intervals
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 interval...
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| Formatua: | Journal article |
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Cambridge University Press
2016
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| _version_ | 1826295672647712768 |
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| author | Teräväinen, J |
| author_facet | Teräväinen, J |
| author_sort | Teräväinen, J |
| collection | OXFORD |
| description | 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 → ∞. |
| first_indexed | 2024-03-07T04:04:39Z |
| format | Journal article |
| id | oxford-uuid:c5be1717-1c47-4927-a96a-8f823d35a2be |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:04:39Z |
| publishDate | 2016 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:c5be1717-1c47-4927-a96a-8f823d35a2be2022-03-27T06:33:09ZAlmost primes in almost all short intervalsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c5be1717-1c47-4927-a96a-8f823d35a2beSymplectic Elements at OxfordCambridge University Press2016Teräväinen, JLet 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 → ∞. |
| spellingShingle | Teräväinen, J Almost primes in almost all short intervals |
| title | Almost primes in almost all short intervals |
| title_full | Almost primes in almost all short intervals |
| title_fullStr | Almost primes in almost all short intervals |
| title_full_unstemmed | Almost primes in almost all short intervals |
| title_short | Almost primes in almost all short intervals |
| title_sort | almost primes in almost all short intervals |
| work_keys_str_mv | AT teravainenj almostprimesinalmostallshortintervals |