Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 117 / 1 / b / Solution

The weighted sifting function isLet ${\lambda }_1=1$ and let the real Selberg sieve weights ${\lambda }_d$ vanish unless $d\mid P_z$ and $d\le D$. Sinceexpansion and the distribution hypothesis giveFor the optimizing Selberg weights, the main quadratic form is $G(D,z{)}^{-1}$, whereThus the general upper bound isIf the sieve level is instead defined as the largest possible least common multiple, one supports the individual weights on $d\le \sqrt{D}$; this is the same statement after replacing $D$ by $\sqrt{D}$.