A conjectured asymptotics giving rise to the desired upper bound for alpha_n

Even though I meant to put an end to this blog publishing a purposed proof of the asymptotic Goldbach conjecture, I finally decided, after the attack that took place in Paris yesterday, to keep on writing new articles here. This is my way to support Charlie Hebdo.

So, I observed that the upper bound $\alpha_{n}\ll \sqrt{n}\log^{2}n$ would follow from both the inequality $r_{0}\ll \log^{4} n$ established in the previous article and the relation $\alpha_{n}\ll \sqrt{nr_{0}(n)}$ that came to my mind rather unexpectedly.

Maybe using once again the inequality of arithmetic and geometric means could help establish the latter rigorously.