Let $g_{n}:=p_{n+1}-p_{n}$ be the $n$- th prime gap, and let's introduce the following summatory functions: $$G_{1}(x):=\sum_{1\leq n\leq x}g_{2n-1}$$ $$G_{2}(x):=\sum_{1\leq n\leq x}g_{2n}$$.

Let's now write $\mathcal{G}(x):=\sum_{k\leq x}\left(\dfrac{G_{2}(k)}{G_{1}(k)}-1\right)$.

Does this last series diverge? Converge towards a positive constant?

Many thanks in advance.