OmegaP S p counts the number of residue classes mod p where at least one polynomial in S vanishes.

The product of degrees of polynomials in a finite set.

The Bateman-Horn constant of a set of polynomials S. This is defined as the infinite product over all primes: $$\frac{1}{D} \prod_p (1 - \frac{1}{p})^{-|S|} (1 - \frac{\omega_p(S)}{p})$$ where $D = \prod_{f \in S} \deg(f)$ is the product of degrees and $\omega_p(S)$ is the number of residue classes mod $p$ where at least one polynomial in $S$ vanishes.

CountSimultaneousPrimes S x counts the number of n ≤ x at which all polynomials in S attain a prime value.

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.