# Goldbach's conjecture and convergence of a Dirichlet series

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).

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 $$1$$?