Let $\theta$ be such that $EH(\theta)$ holds, where $EH$ stands for Elliott-Halberstam. Can one get an explicit lower bound $\delta_{\theta}$ for the quantity $\delta$ appearing in the weak Hardy-Littlewood-Goldbach stated in https://www.sciencedirect.com/science/article/abs/pii/S0022314X21002031? In particular, would Elliott-Halberstam conjecture imply the weak Hardy-Littlewood-Goldbach conjecture for some $\delta$ close to $1$?
Indeed it is known that the greater $\theta$ is, the smaller the quantity $l_{k}(\theta):=\lim\inf p_{n+k}-p_{n}$ gets. Define $r_{0}(m)$ as the smallest non negative integer $r$ such that both $m-r$ and $m+r$ are prime and $k_{0}(m):=\pi(m+r_{0}(m))-\pi(m-r_{0}(m))$. I call such an integer $r$ a primality radius of $m$. Say also that an integer $m$ is $k$-central if $k_{0}(m)=k$ and denote by $\mathbb{N}_{k}$ the set of $k$-central integers.
Then one has $l_{k}(\theta)=2\lim\inf_{m\in\mathbb{N}_{k}}r_{0}(m)$, so the greater $\theta$ is, the smaller $r_{0}$ and very likely the more numerous primality radii of an integer of given magnitude get.
