Uniform Hardy–Littlewood conjecture for shifted prime correlations

Let $\Lambda$ be the von Mangoldt function, define the singular series

$$\mathfrak S=2\prod_{p>2}\left(1-\frac1{(p-1)^2}\right),$$

and for $h\ne0$ let $\mathfrak S(h)=0$ when $h$ is odd, while for even $h$ let

$$\mathfrak S(h)=\mathfrak S\prod_{\substack{p\mid h\p>2}}\frac{p-1}{p-2}.$$

Set $\mathfrak S_n(h)=\delta((n,h)=1)\mathfrak S(nh)$. Uniform Hardy–Littlewood conjecture. For every $\varepsilon>0$,

$$\sum_{m\leq x}\Lambda\left(\frac mn\right)\Lambda(m\pm h)=\frac{\mathfrak S_n(h)}n x+O_\varepsilon(x^{1/2+\varepsilon})$$

uniformly for $1\leq h,n\leq x^{1-\varepsilon}$.

This is a uniform prime-pair correlation estimate, used in the paper to control the larger-$\alpha$ range of the twisted pair correlation. The source presents it as a Hardy–Littlewood-type hypothesis rather than a proved result.