Bateman-Horn Conjecture

The Bateman-Horn Conjecture* Given a finite collection of distinct irreducible polynomials non-constant $f_1, f_2, \dots, f_k \in \mathbb{Z}[x]$ with positive leading coefficients that satisfy the Schinzel condition, the number of positive integers n ≤ x for which all polynomials $f_i$ are simultaneously prime is asymptotic to: $C(f_1, f_2, \dots, f_k) x / (log x)^k$ where $C$ is the Bateman-Horn constant given by the convergent infinite product: $C = \frac{1}{D}\prod_{p\in\mathbb{P}} (1 - 1/p)^(-k) · (1 - \omega_p/p)$ Here $\omega_p/p$ is the number of residue classes modulo $p$ for which at least one polynomial vanishes.

The Schinzel condition ensures that for each prime $p$, there exists some integer $n$ such that $p$ does not divide the product $f_(n) f_2(n) \dotsb f_(n)$, which guarantees the infinite product converges to a positive value.