A1 Refereed original research article in a scientific journal
Almost primes in almost all short intervals
Authors: Teravainen J
Publisher: CAMBRIDGE UNIV PRESS
Publication year: 2016
Journal: Mathematical Proceedings of the Cambridge Philosophical Society
Journal name in source: MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY
Journal acronym: MATH PROC CAMBRIDGE
Volume: 161
Issue: 2
First page : 247
Last page: 281
Number of pages: 35
ISSN: 0305-0041
eISSN: 1469-8064
DOI: https://doi.org/10.1017/S0305004116000232
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/17473466
Let E-k be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log(1+epsilon) x] contain E-3 numbers, and almost all intervals [x, x + log(3.51) x] contain E-2 numbers. By this we mean that there are only 0(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 epsilon in the exponent. The theorem on E-2 numbers improves a result of Harman, which had the exponent 7+epsilon in place of 3.51. We also consider general E-k numbers, and find them on intervals whose lengths approach log x as k -> infinity.
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