Theorem (Sieve of Erastothenes – Legendre). Assuming that:

Then

\begin{array}{l} S (A, P, z) \\ = | A | \prod_{\begin{array}{c} p \leq 2 \\ p \in P \end{array}} (1 - g (p)) + O (x^{\frac{1}{2}} {(\log x)}^{\frac{1}{2}} 2^{- \frac{\log x}{4 \log z}} {(\sum_{\begin{array}{c} d \leq x \\ d | P (z) \end{array}} | R_{d} |^{2})}^{\frac{1}{2}} + | A | e^{\frac{\log x}{\log z}} \prod_{\begin{array}{c} p \leq z \\ p \in P \end{array}} {(1 + g (p))}^{e}) \end{array}