Generalized Hardy–Littlewood conjecture for affine prime patterns

Fix $R\in\mathbb{N}$ and $C>0$. Let $a_i,b_i\in\mathbb{Z}$ satisfy $(a_i,b_i)=1$ and $a_i>0$ for $1\leq i\leq R$. Define the Hardy–Littlewood singular series

$$\mathfrak{S}=\prod_{p\ \mathrm{prime}}\left(1-\frac{\rho(p)}p\right)\left(1-\frac1p\right)^{-R},$$

where $\rho(p)$ is the number of solutions of $\prod_{i=1}^R(a_i n+b_i)=0$ in $\mathbb{Z}/p\mathbb{Z}$. Generalized Hardy–Littlewood conjecture. If this product converges, then

$$\left|\left\{n\leq x:\ \forall i,\ a_i n+b_i\ \text{is prime}\right\}\right|=(1+o_{R,C}(1))\mathfrak{S}\frac{x}{(\log x)^R}+o_{R,C}\left(\frac{x}{(\log x)^R}\right),$$

for $x$ satisfying $(\log x)^C\geq |a_i|,|b_i|$ for $1\leq i\leq R$. This is a quantitative generalization of the Hardy–Littlewood $k$-tuples conjecture and is related to the generalized Hardy–Littlewood conjecture of Green and Tao; the uniformity in the coefficients is the feature emphasized by the authors, and no resolution is supplied.