= Fundamental lemma of sieve theory
The fundamental lemma of sieve theory says that upper and lower <sieve theory> estimates approach the expected local-density product when the available distribution level is a sufficiently large power of the sifting limit. One formulation uses a finite set $A$ of positive integers and
$$
|\{n\in A:d\mid n\}|=Xg(d)+r_d,\qquad V(z)=\prod_{p<z}(1-g(p)),\qquad s=\frac{\log D}{\log z}.
$$
Here $d$ ranges over <squarefree integers> up to $D$, $g$ is <multiplicative> with $0\leq g(p)<1$, and the local densities satisfy a fixed-dimension bound
$$
\frac{V(w)}{V(z)}\leq C\left(\frac{\log z}{\log w}\right)^\kappa\quad(2\leq w\leq z).
$$
Then a standard version yields
$$
|\{n\in A:\gcd(n,P(z))=1\}|=XV(z)(1+O_{\kappa,C}(e^{-s}))+O\!\left(\sum_{\substack{d\leq D\\d\text{ squarefree}}}|r_d|\right),\qquad P(z)=\prod_{p<z}p.
$$
The <fundamental lemma of sieve theory> is useful only after both errors are controlled: large $s$ reduces the relative main-term error, while the summed remainders measure the actual arithmetic distribution. It then guarantees survivors, often <almost primes>, without claiming they are prime.<Some sieve theory by Terence Tao (254A Notes 4)>[Equation (22), Lemma 17 and Corollary 19; endpoint convention uses primes below z]{ref}
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