This question is a follow-up to About Goldbach's conjecture.

I would like to know if an unconditional upper bound for $\alpha_{n}$ can be obtained from the error tem in Mertens' third theorem which, as stated in the French wikipedia, says that $\prod_{p\leq n}(1-\frac{1}{p})=\frac{e^{-\gamma}}{\log n}(1+O(\frac{1}{\log n}))$ for $n\geq 2$.

This question is a follow-up to About Goldbach's conjecture.

I would like to know if an unconditional upper bound for $\alpha_{n}$, defined as $n(N_{2}(n)-\dfrac{nN_{1}(n)}{P(n)})$ (where $N_{2}(n):=\sharp\{0\leq r<n-p_{\pi(\sqrt{2n-3})},(n-r,n+r)\in\mathbb{P}^{2}\}$, $N_{1}(n)$ is the number of integers $m$ below $P(n)=\prod_{p\leq \sqrt{2n-3}}p$ such that $m\pm r\not\equiv 0\pmod p$ for all prime $p$ below $\sqrt{2n-3}$) can be obtained from the error tem in Mertens' third theorem which, as stated in the French wikipedia, says that $\prod_{p\leq n}(1-\frac{1}{p})=\frac{e^{-\gamma}}{\log n}(1+O(\frac{1}{\log n}))$ for $n\geq 2$.