Theorem 2.2 (Selberg sieve). Assuming that:

$z \geq 2$
$A \subseteq ℤ$ finite
$P \subseteq ℙ$
Assume the sieve hypothesis
$h : ℕ \to [0, \infty)$ be the multiplicative function supported on square-free numbers, given on the primes by

$h (p) = {\begin{matrix} \frac{g (p)}{1 - g (p)} & p \in P \\ 0 & p \notin P \end{matrix}$

Then

S (A, P, z) \leq \frac{| A |}{\sum_{d \leq z} h (d)} + \sum_{\begin{array}{c} d \leq z^{2} \\ d | P (z) \end{array}} τ_{3} (d) | R_{d} | .