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Dense clusters of primes in subsets

Published online by Cambridge University Press:  01 April 2016

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 $p_{n+m}-p_{n}\leqslant {\it\epsilon}\log x$.

Type
Research Article
Copyright
© The Author 2016 

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