When the sieve works II

For a set of primes P, let Ψ(x;P) be the number of positive integers n≤x all of whose prime factors lie in P. In this paper we classify the sets of primes P such that Ψ(x;P) is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matomäki [A. Granville, D. Koukoulopoulos and K. Matomäki, When the sieve works, Duke Math. J. 164 2015, 10, 1935–1969] and their main conjecture is proved in this paper. In particular, our main theorem implies that, if not too many large primes are sieved out in the sense that

∑p∈Px1/v0 and v≥u≥1, then Ψ(x;P)≫ε,vx∏p≤xp∉P(1−1p).