This question is related to https://mathoverflow.net/questions/61842/about-goldbachs-conjecture
Let $ y(x) $ be a function such that $ \alpha_{n}=O(y(n)) $ . taking $ y(x) : =x^{1/2}\log^{2} x $ leads to $ y'=\dfrac{1}{2}x^{-1/2}\log^{2}x+2x^{-1/2}\log x $ . As $1/ \log x $ is roughly equal to $ \pi(x)/x $ , is there any reason to believe that $ \alpha_{n}=O(z(n)) $ where the function $ z(x) $ fulfills the differential equation $ z'=\dfrac{z}{2\Delta}+z\dfrac{\pi(x+\Delta)-\pi(x-\Delta)}{\Delta} $ where $ \Delta : =\Delta(x) $ is such that $ \Delta(x)\leq (1+o(1))x $?