{c}
{wiki=Selberg_sieve}

Suppose $|\mathcal A_d|=Xg(d)+r_d$ for $d\mid P(z)$. If $\lambda_1=1$ and the real Selberg weights $\lambda_d$ vanish unless $d\mid P(z)$ and $d\leq D$, then
$$
S(\mathcal A,\mathcal P,z)
\leq X\sum_{d,e}\lambda_d\lambda_e g([d,e])
+\sum_{d,e}\lambda_d\lambda_e r_{[d,e]}.
$$
Optimizing the main quadratic form gives $X/G(D,z)$, where
$$
G(D,z)=\sum_{\substack{\ell\leq D\\\ell\mid P(z)}}
\prod_{p\mid\ell}\frac{g(p)}{1-g(p)},
$$
up to the harmless replacement of $D$ by $D^2$ under the alternative convention that $D$ denotes the level of the least common multiples.


 Selberg upper-bound sieve

ID: selberg-upper-bound-sieve