For a set of primes $𝒫$, let $\Psi (x;𝒫)$ be the number of positive integers $n\le x$ all of whose prime factors lie in $𝒫$. In this paper we classify the sets of primes $𝒫$ such that $\Psi (x;𝒫)$ 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

中文翻译：

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筛子何时工作II

对于一组素数 $𝒫$，让 $\Psi （X;𝒫）$ 是正整数的数量 $ñ\le X$ 所有主要因素都在于 $𝒫$。在本文中，我们对素数集进行分类$𝒫$ 这样 $\Psi （X;𝒫）$在其预期值的恒定因子之内。这项任务最近由Granville，Koukoulopoulos和Matomäki[A. Granville，D。Koukoulopoulos和K.Matomäki，当筛子起作用时，Duke Math。[J. 164 2015，10，1935–1969]及其主要猜想在本文中得到证明。特别地，我们的主定理意味着，如果没有太多的大质数被筛除，就意味着