Sifting function ($S(A,\mathcal{P},z)$)

For a finite integer set $A$ and a set of primes $\mathcal{P}$, the sifting function counts elements of $A$ divisible by no prime in $\mathcal{P}$ below $z$:

For a nonnegative weighted sequence $\mathcal{A}=(a_n)$, the same notation means $S(\mathcal{A},\mathcal{P},z)={\sum }_{(n,P(z))=1}{a}_n$.

Put $|{\mathcal{A}}_d|={\sum }_{d\mid n}{a}_n$. A nonnegative sequence has distribution $(X,P(z),g,(r_d))$ when $g$ is multiplicative on the squarefree divisors of $P(z)$, $0\le g(p)<1$, andfor every $d\mid P(z)$.

If $w<z$, ordering the least prime factor giveswhere $({\mathcal{A}}_p{)}_n=a_{pn}$ in a set formulation, or equivalently ${\mathcal{A}}_p$ restricts the original sequence to terms divisible by $p$.