Dense clusters of primes in subsets. (1st April 2016)

• Record Type: Journal Article
• Authors: Maynard, James
• Is Part Of: Compositio mathematica. Volume 152:Number 7(2016:Jul.)
• Page Start: 1517
• Page End: 1554
• Publication Date: 2016-04-01
• Subjects: 11N05, -- 11N35 (primary); prime numbers, -- sieve methods; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X16007296
• Languages: English
• ISSNs: 0010-437X

Abstract

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