Quantitative Hardy–Littlewood prime tuples conjecture

Let $\mathcal{P}$ denote the set of primes, and let $1_{\mathcal{P}}$ be its indicator function. For a tuple of distinct integers ${\mathcal H}=\{h_1,\dots,h_k\}$, let ${\mathfrak S}({\mathcal H})$ denote its singular series. Quantitative Hardy–Littlewood prime tuples conjecture. There exist two absolute constants $\varepsilon>0$ and $C>0$ such that, for all $x\geq 10$, all $k\leq(\log\log x)^5$, and all tuples ${\mathcal H}=\{h_1,\dots,h_k\}\subset[0,\log^2 x]$ of distinct integers $h_1,\dots,h_k$, one has

$$\left|\sum_{n\leq x}1_{\mathcal{P}}(n+h_1)\dots 1_{\mathcal{P}}(n+h_k)-{\mathfrak S}({\mathcal H})\int_2^x\frac{dy}{\log^k y}\right|\leq Cx^{1-\varepsilon}.$$

This is a quantitative strengthening of the Hardy–Littlewood prime tuples conjecture, used in the paper to establish convergence of Erdős's alternating series assuming the conjecture.