The sieve problem: estimate $S(\mathcal{A}, \mathcal{P}, z) = |\{a \in \mathcal{A} : (a, P(z)) = 1\}|$ where $\mathcal{A}$ is a finite set of integers, $\mathcal{P}$ is a set of primes, and $P(z) = \prod_{p \in \mathcal{P}, p < z} p$. The sifted set consists of elements of $\mathcal{A}$ not divisible by any $p \in \mathcal{P}$ with $p < z$. By inclusion-exclusion: $S(\mathcal{A}, \mathcal{P}, z) = \sum_{d | P(z)} \mu(d) |\mathcal{A}_d|$ where $\mathcal{A}_d = \{a \in \mathcal{A} : d | a\}$.

Write $|\mathcal{A}_d| = \frac{\omega(d)}{d} X + r_d$ where $X = |\mathcal{A}|$, $\omega$ is a multiplicative function (the sieve density), and $r_d$ is the remainder term. The sieve dimension $\kappa$ is defined by $\sum_{p < z} \frac{\omega(p)}{p} \approx \kappa \log\log z$. For primes: $\kappa = 1$ (linear sieve). For twin primes: $\kappa = 2$ (2-dimensional sieve). The dimension governs the difficulty: $\kappa = 1$ is the easiest, higher $\kappa$ gives weaker results.