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

D. H. J. Polymath

For any $m\ge 1$, let $H_m$ denote the quantity ${\mathrm{liminf}}_{n\to \infty }(p_{n+m}-p_n)$. A celebrated recent result of Zhang showed the finiteness of $H_1$, with the explicit boun…