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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