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

For a positive integer $d$, defineThe sequence $\mathcal{A}$ has sieve distribution $(X,P_z,g,(r_d))$ when $g$ is a multiplicative arithmetic function on the squarefree divisors of $P_z$, with $0\le g(p)<1$, andfor every $d\mid P_z$. The function $g(d)$ models the local density of divisibility by $d$, while $r_d$ is the remainder.