Multivariable Bateman–Horn conjecture

Let $f_1,\ldots,f_r\in\mathbb{Z}[x_1,\ldots,x_n]$ be irreducible polynomials such that $\prod_{i=1}^r f_i$ has no repeated polynomial factors. Let $\mathcal{B}\subset\mathbb{R}^n$ be a non-empty box such that, writing $f_{i0}$ for the top-degree part of $f_i$, one has

$$f_{i0}(\mathcal{B})\subset(1,\infty)\qquad(1\leqslant i\leqslant r).$$

For $P>0$, let $\pi_{f_1,\ldots,f_r}(P\mathcal{B})$ be the cardinality of the set of integer vectors $\mathbf{x}\in\mathbb{Z}^n\cap P\mathcal{B}$ for which every $f_1(\mathbf{x}),\ldots,f_r(\mathbf{x})$ is a positive prime number.

Multivariable Bateman–Horn conjecture. As $P\to+\infty$,

$$\pi_{f_1,\ldots,f_r}(P\mathcal{B})\sim\left(\prod_{p\ \text{prime}}\frac{1-p^{-n}\#\{\mathbf{x}\in\mathbb{F}_p^n:f_1(\mathbf{x})\cdots f_r(\mathbf{x})=0\}}{(1-1/p)^r}\right)\int_{P\mathcal{B}}\frac{\mathrm{d}\mathbf{x}}{\prod_{i=1}^r\log f_{i0}(\mathbf{x})}.$$

This is the proposed extension of the Bateman–Horn heuristic from one variable to arbitrarily many variables. It predicts the asymptotic density of simultaneous prime values in expanding boxes and is open in general.