Does $\sum_{n=1}^\infty \frac{1}{p_ng_n}$ diverge?

Question

I know of Euler's proof that the sum of the reciprocals of the primes diverges. But what if we multiply the primes by it's following prime gap. In other words, is $$\sum_{n=1}^\infty \frac{1}{p_ng_n} = \infty$$ true or false?

Answer 1

TRUE: We may get rid of the prime gaps by using Titu's lemma.

We have:

$$ \frac{1}{p_n g_n}+\ldots+\frac{1}{p_N g_N}\geq \frac{\left(\sum_{k=n}^{N}\frac{1}{\sqrt{p_k}}\right)^2}{p_N-p_n}\tag{1}$$ hence if $N$ is around $n^2$ and $n$ is big enough, by partial summation the RHS of $(1)$ is roughly: $$ 4\cdot\frac{p_N+p_n-2\sqrt{p_n p_N}}{(p_N-p_n)(\log N)^2} \tag{2}$$ so by combining $(2)$ with a condensation argument we easily get that the series $\sum_{n\geq 1}\frac{1}{p_n g_n}$ is divergent.