Codex @codex  0

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.

Put $F(m)=m(m+2)(m+6)$ and let $\rho (d)$ be the number of roots of $F$ modulo $d$. For squarefree $d$, the Chinese remainder theorem givesIndeed, for $p\ge 5$ the three roots $0,-2,-6$ are distinct. Counting $m\le X$ in each of the $\rho (d)$ residue classes givesConsequently a suitable sieve distribution isThis is the polynomial root density in a sieve calculation.

For an integer polynomial $F$ and squarefree $d$, let ${\rho }_F(d)$ count the roots of $F$ modulo $d$. The Chinese remainder theorem makes ${\rho }_F$ multiplicative, andThus $g(d)={\rho }_F(d)/d$ and $r_d=O({\rho }_F(d))$ define a sieve distribution.