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 $n\geq 2$.