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Dense clusters of primes in subsets

Part of: Multiplicative number theory

Published online by Cambridge University Press:  01 April 2016

James Maynard
James Maynard*
Affiliation:
Centre de recherches mathématiques, Université de Montréal, Pavillon André-Aisenstadt, 2920 Chemin de la tour, Room 5357, Montréal (Québec), Canada H3T 1J4 email james.alexander.maynard@gmail.com Current address:Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX1 6GG, UK
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Abstract

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We prove a generalization of the author’s work to show that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length $(\log x)^{{\it\epsilon}}$ containing $\gg _{{\it\epsilon}}\log \log x$ primes, and show lower bounds of the correct order of magnitude for the number of strings of $m$ congruent primes with $p_{n+m}-p_{n}\leqslant {\it\epsilon}\log x$.

Keywords

prime numberssieve methods

MSC classification

Primary: 11N05: Distribution of primes 11N35: Sieves
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 7 , July 2016 , pp. 1517 - 1554
Copyright
© The Author 2016 

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