Are prime gaps of even index essentially larger than those of odd index?

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?

• Do we even know the property (weaker than the statement that the sum $\mathcal{G}(x)$ converges) that $\lim_{k\to \infty}\frac{G_2(k)}{G_1(k)} = 1$? – Mark Fischler Feb 4 '16 at 19:48