Bateman-Horn Conjecture: 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 $n \leq x$ for which all polynomials $f_i$ are simultaneously prime is asymptotic to: $\frac{C}{D} \frac{x}{(\log x)^k}$ where $D = \prod_i \deg(f_i)$ and $C$ is the Bateman-Horn constant given by the convergent infinite product: $C = \prod_{p\in\mathbb{P}} (1 - 1/p)^{-k} (1 - \omega_p/p)$ Here $\omega_p$ is the number of residue classes modulo $p$ for which at least one polynomial vanishes. The product is only conditionally convergent: it is the limit as $N \to \infty$ of the partial products over the primes $p < N$.

The Schinzel condition ensures that for each prime $p$, there exists some integer $n$ such that $p$ does not divide the product $f_1(n) f_2(n) \dotsb f_k(n)$, which guarantees the infinite product converges to a positive value.