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Under Goldbach's conjecture, denote by $ r_{0}(n)=\inf\{r\gt 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ and by $ k_{0}(n)=\pi(n+r_{0}(n))-\pi(n-r_{0}(n)) $ for $ n $ large enough. Say $ n $ is $ k $ -central if and only if $ k_{0}(n)=k $.

Let's define $ \delta_{k}^{+} : =\lim\sup\dfrac{\log r_{0}(n)}{\log\log n} $ when $n $ runs through $ k $ -central numbers and tends to $ \infty $, $ \delta_{k}^{-} $ similarly with this time a $ \lim\inf $, and $ \delta_{k} $ as the limit as $ n $ tends to $ \infty $ of $\dfrac{1}{\pi_{k,C}(n)} \sum_{C<m\leq n,k_{0}(m)=k}\dfrac{\log r_{0}(m)}{\log\log m} $ for $ C $ an absolute positive constant, where $\pi_{k,C}(x) $ is the number of $ k $ -central numbers greater than $ C $ not exceeding $ x $. Does one have $ \delta_{k}=\frac{\delta_{k}^{+}+\delta_{k}^{-}}{2} $ for all $ k>0$? Are those quantities actually independent on $ k $? Is there a good reason to believe that for all $ k>0 $ one has $ \delta_{k}^{+}=2 $?

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