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Almost primes in almost all short intervals

Published online by Cambridge University Press:  13 April 2016

JONI TERÄVÄINEN
JONI TERÄVÄINEN*
Affiliation:
Department of Mathematics, University of Turku, 20014 Turku, Finland. e-mail: joni.p.teravainen@utu.fi
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Abstract

Let Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵx] contain E3 numbers, and almost all intervals [x,x + log3.51x] contain E2 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 E3 numbers is optimal up to the ϵ in the exponent. The theorem on E2 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 → ∞.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 2 , September 2016 , pp. 247 - 281
Copyright
Copyright © Cambridge Philosophical Society 2016 

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