Bateman–Horn conjecture for admissible irreducible integer polynomials

Let $f_1,\ldots,f_k$ be irreducible integer polynomials satisfying the admissibility condition. Let $P(x)$ denote the number of integers $n\leq x$ for which all $f_i(n)$ are prime. Bateman–Horn conjecture. The count is asymptotically

$$P(x)\sim \frac{C}{\prod \deg f_i}\int_2^x\frac{dt}{(\log t)^k},$$

where $C$ is a product over primes adjusted for local obstructions. This conjecture predicts the density of simultaneous prime values of admissible polynomial families and underlies the paper’s discussion of prime-rich polynomials; it remains open in general.