The modified Bateman–Horn conjecture

Let $f_{1},f_{2},\cdots,f_{M}\in\mathbb{Z}[x]$ be distinct irreducible polynomials with positive leading coefficients. Define

$$\pi(f_{1},f_{2},\cdots,f_{M};x)=\#\{n\leq x:f_{1}(n),f_{2}(n),\cdots,f_{M}(n)\text{ are primes}\}.$$

Put $f(n)=\prod_{i=1}^{M}f_{i}(n)$, suppose that $f(n)$ does not vanish identically modulo any prime, and let $n_{0}$ be the smallest integer such that, for every $n>n_{0}$, $f_{i}(n)>1$ for $1\leq i\leq M$. Let $C(f_{1},f_{2},\cdots,f_{M})$ be the Bateman–Horn constant

$$C(f_{1},f_{2},\cdots,f_{M})=\prod_{p}\left(1-\frac{1}{p}\right)^{-M}\left(1-\frac{\omega_{f}(p)}{p}\right),$$

where $\omega_{f}(p)$ is the number of solutions of $f(n)\equiv0\pmod p$. Modified Bateman–Horn conjecture. For large values of $x$,

$$\pi(f_{1},f_{2},\cdots,f_{M};x)\sim C(f_{1},f_{2},\cdots,f_{M})\int_{n_{0}}^{x}\frac{dt}{\prod_{i=1}^{M}\log f_{i}(t)}.$$

This modification retains the local correction factor while replacing the degree-based logarithmic approximation with the actual quantities $\log f_{i}(t)$, thereby addressing deviations for non-monic polynomials. The proposed asymptotic is open, and the paper presents empirical evidence for its accuracy.