We are sieving a set of size X (perhaps the integers in an interval) with the primes for a given set P. The "probability" that a given element of our set is divisible by p, from P, is about 1/p. In order to use some sort of inclusion-exclusion argument, we will need to know the "probability" that a given element of our set is divisible by pq, with p,q from P. We expect this to be 1/pq, but if pq>X then this will have to rather inaccurate. So the many wonderful results of sieve theory typically work under the assumption the primes in P are less than X^(1/2).
But what if we allow some of the primes in P to be greater than X^(1/2) ? We know many examples where the number of integers left unsieved is far less than one might guess, in this case. In this article Dimitris Koukoulopoulos, Kaisa Matomaki and I show that there exists a constant k>1 such that if we are sieving the interval [1,X], and the sum of the reciprocals of the primes up to X that are not in P, is >k, then the number of integers left unsieved is roughly as one might guess. Moreover we conjecture that one can take any k>1, and speculate that an analogous result may be true when sieving any interval.
The proof revolves around a quantative estimate for additive combinatorics for sumsets.
Don't be seduced by the zeros!
Different approaches to the distribution of primes
The Princeton Companion to Mathematics: Analytic number theory
Prime number patterns (2009 Ford Prize)
It is easy to determine whether a given integer is prime (2008 Chauvenet Prize)
Prime number races (2007 Ford Prize)
It's as easy as abc
Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle (1995 Hasse Prize)