Definition (Sieve problem). Let $P\subseteq \mathbb{P}$. Let

$$P(z)=\prod _{\begin{array}{c}p\in P\\ p\le z\end{array}}p,$$

and let $A\subseteq \mathbb{Z}$. Denote

$$S(A,P,z)=|\{n\in A:(n,P(z))=1\}|.$$