Dense clusters of primes in subsets

By James Maynard. Compositio Mathematica, 2014

94 citations recorded by the source corpus.

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

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/98992782083487597c3f1f5428d9ad91c88c6a6a

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Long gaps between primes (2014, methodology): “The proof of this theorem will rely on the estimates for multidimensional prime-detecting sieves established by the fourth author in [28], and will be the focus of subsequent sections.”
Variants of the Selberg sieve, and bounded intervals containing many primes (2014, unknown): “ ⁄ „ p 1 1 p1 1{logx O 1 p2 !exp ‚ p 1 p1 1{logx ⁄expploglogx Op1qq !logx: Thus Kp˘ 1;:::;˘k;˘1;:::;˘1 k qOplog 3kxq: Combining this with the rapid decrease of fj;gj, we see that the contribution to (38) outside of the cube tmaxp˘ 1;:::;˘k;˘1;:::;˘1 k q⁄ ? logxu(say) is negligible. Thus it will suce”