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Assuming Goldbach's conjecture, let's denote by $ r_{0}(n) : =\inf\{r>0, (n-r,n+r)\in\mathbb{P}^{2}\} $. The assumption of GC implies $ r_{0}(n)<n $.

Let's now consider the series $ G(s) : =\sum_{n>0}\frac{a_{n}}{n^s} $ with $ a_{n}=r_{0}(n) $ if $ n $ is composite and $ a_{n}=1 $ otherwise.

Unconditionnally, $ G(s) $ converges for $\Re(s)>2 $. What would be the infimum of the exponents $\beta $ such that $ r_{0}(n)\ll n^{\beta} $ if we could prove the abscissa of convergence of $ G(s) $ is $ 1 $?

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