The linear sieve is a powerful tool to tackle problems related to the primes, when combined with equidistribution estimates for the remainder. In 1977 Iwaniec introduced a well-factorable modification of the linear sieve to prove there are infinitely many integers n such that n2+1 has at most two prime factors. Furthermore, the (well-factorable) linear sieve leads to the best known upper bounds for twin primes. These bounds use work of Bombieri, Friedlander, and Iwaniec from 1986, showing these sieve weights equidistribute primes of size x in arithmetic progressions to moduli up to x4/7. This level was recently increased to x7/12 by Maynard.We introduce a new modification of the linear sieve whose weights equidistribute primes of size x to level x10/17. As an application we refine a 2004 upper bound for twin primes of Wu, which gives the largest percent improvement since the work of Bombieri, Friedlander, and Iwaniec.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
