# Variants of the Selberg sieve, and bounded intervals containing many primes

```@article{Polymath2014VariantsOT,
title={Variants of the Selberg sieve, and bounded intervals containing many primes},
author={D. H. J. Polymath},
journal={Research in the Mathematical Sciences},
year={2014},
volume={1},
pages={1-83}
}```
• D. Polymath
• Published 18 July 2014
• Mathematics
• Research in the Mathematical Sciences
For any m≥1, let Hm denote the quantity liminfn→∞(pn+m−pn). A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the Polymath8 project) to H1≤4680, and then by Maynard to H1≤600, who also established for the first time a finiteness result for Hm for m≥2, and specifically that Hm≪m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1≤12, improving upon the previous bound H1≤16 of…
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In a recent and much celebrated breakthrough, Maynard and Tao have independently proved a certain approximation to the prime k-tuple conjecture. We have subsequently seen numerous interesting
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Erd\"os conjectured that the set J of limit points of d_n/logn contains all nonnegative numbers, where d_n denotes the nth primegap. The author proved a year ago (arXiv: 1305.6289) that J contains an
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