Small gaps between primes
Annals of mathematics, Tome 181 (2015) no. 1, pp. 383-413
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We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, $\liminf_{n}(p_{n+m}-p_n)<\infty$ for every integer $m$. We also show that $\liminf(p_{n+1}-p_n)\le 600$ and, if we assume the Elliott-Halberstam conjecture, that $\liminf_n(p_{n+1}-p_n)\le 12$ and $\liminf_n (p_{n+2}-p_n)\le 600$.
@article{10_4007_annals_2015_181_1_7,
author = {James Maynard},
title = {Small gaps between primes},
journal = {Annals of mathematics},
pages = {383--413},
publisher = {mathdoc},
volume = {181},
number = {1},
year = {2015},
doi = {10.4007/annals.2015.181.1.7},
mrnumber = {3272929},
zbl = {1306.11073},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.181.1.7/}
}
James Maynard. Small gaps between primes. Annals of mathematics, Tome 181 (2015) no. 1, pp. 383-413. doi: 10.4007/annals.2015.181.1.7
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