Under Goldbach's conjecture, let's denote for $n$ a positive integer greater than $1$ by $\mathbb{G}(n)$ the set of positive integers $r$ such that both $n-r$ and $n+r$ are prime.
Denoting by $S(x):=\sum_{n\le x}\sum_{r\in\mathbb{G}(n)}\frac{1}{r}$, is there a non negative real number $\alpha$ such that $S(x)\sim\dfrac{x}{\log^{1-\alpha}x.(\log\log x)^{\alpha}}$? If so, what is the value of $\alpha$?
A few numerical computations seem to suggest one may take $\alpha=\frac{\sqrt{5}-1}{2}$.
Edit: actually the assumption of Goldbach's conjecture is unnecessary. Were it false, there would be integers $n$ such that $\mathbb{G}(n)$ would be empty, making the inner sum equal $0$ for those $n$. As we know that the number of even integers below $x$ that are not Goldbach numbers is an $o(x)$, that doesn't prevent $S(x)$ from existing and being positive. One may even say that the truth of Goldbach's conjecture would be a consequence of the maximality of this sum among its possible values.
I used Gauss' method of least squares to get the best fit for different values of $\alpha$. It seems it is attained for $\alpha$ close to $2/3$:
This would mean that, on average, $R(n):=\sum_{r\in\mathbb{G}(n)}\frac{1}{r}$ is greater than $\dfrac{1}{\log n}$. So if there are few integers $r>0$ such that $n-r$ and $n+r$ are prime, $r_{0}(n)$ defined as the smallest such $r$ can't be too large.