Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 150 / 2 / a / Solution

Let $\mathcal{A}=(a_n)$ be a finite nonnegative sequence, let $\mathcal{P}$ be a set of prime numbers, and defineThe sifting function isFor a finite set $A$ of integers, take $a_n$ to be the number of occurrences of $n$ in $A$; then $S(A,\mathcal{P};z)$ counts members divisible by no $p\in \mathcal{P}$ below $z$. For $d\mid P(z)$, write