A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä

Almost primes in almost all short intervals




TekijätTeravainen J

KustantajaCAMBRIDGE UNIV PRESS

Julkaisuvuosi2016

JournalMathematical Proceedings of the Cambridge Philosophical Society

Tietokannassa oleva lehden nimiMATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY

Lehden akronyymiMATH PROC CAMBRIDGE

Vuosikerta161

Numero2

Aloitussivu247

Lopetussivu281

Sivujen määrä35

ISSN0305-0041

eISSN1469-8064

DOIhttps://doi.org/10.1017/S0305004116000232

Rinnakkaistallenteen osoitehttps://research.utu.fi/converis/portal/detail/Publication/17473466


Tiivistelmä
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.

Ladattava julkaisu

This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail. Please cite the original version.





Last updated on 2024-26-11 at 11:51