The weighted <sifting function> is
$$
S(\mathcal A,\mathcal P,z)
=\sum_{\substack{n\geq1\\(n,P_z)=1}}a_n.
$$
Let $\lambda_1=1$ and let the real <Selberg upper-bound sieve>[Selberg sieve weights] $\lambda_d$ vanish unless $d\mid P_z$ and $d\leq D$. Since
$$
1_{(n,P_z)=1}\leq\left(\sum_{d\mid(n,P_z)}\lambda_d\right)^2,
$$
expansion and the distribution hypothesis give
$$
S(\mathcal A,\mathcal P,z)
\leq X\sum_{d,e}\lambda_d\lambda_e g([d,e])
+\sum_{d,e}\lambda_d\lambda_e r_{[d,e]}.
$$
For the optimizing Selberg weights, the main quadratic form is $G(D,z)^{-1}$, where
$$
G(D,z)=\sum_{\substack{\ell\leq D\\\ell\mid P_z}}
\prod_{p\mid\ell}\frac{g(p)}{1-g(p)}.
$$
Thus the general upper bound is
$$
\boxed{
S(\mathcal A,\mathcal P,z)
\leq\frac{X}{G(D,z)}
+\sum_{\substack{d,e\mid P_z\\d,e\leq D}}
|\lambda_d\lambda_e r_{[d,e]}|.}
$$
If the sieve level is instead defined as the largest possible least common multiple, one supports the individual weights on $d\leq\sqrt D$; this is the same statement after replacing $D$ by $\sqrt D$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-117/1/b/solution