# 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} }

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… Expand

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