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Large prime factors on short intervals

Part of: Multiplicative number theory

Published online by Cambridge University Press:  05 September 2019

JORI MERIKOSKI
JORI MERIKOSKI*
Affiliation:
Department of Mathematics, University of Turku, FI-20014 University of Turku, Finland e-mail: jori.e.merikoski@utu.fi
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Abstract

We show that for all large enough x the interval [x, x + x1/2 log1.39x] contains numbers with a prime factor p > x18/19. Our work builds on the previous works of Heath–Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals [x, x + x1/2 + ϵ]. We also incorporate some ideas from Harman’s book Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomäki and Radziwiłł (2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of log x when applying Harman’s sieve method.

MSC classification

Primary: 11N05: Distribution of primes
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 1 , January 2021 , pp. 1 - 50
Copyright
© Cambridge Philosophical Society 2019

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