Please help me evaluate the following infinite sum: $\sum\limits_{k=1}^{\infty}\frac{1}{p_k(p_k-1)}$

Question

I'm trying to evaluate:

$\displaystyle{\sum_{k=1}^{\infty}\frac{1}{p_k(p_k-1)}}$, where $p_k$ is the $k$-th prime number.

But I cannot even figure out how to begin. I have a feeling that this could involve the prime zeta function, but I'm not sure. In fact, I'm not even sure an analytic closed-form solution is possible. Could you please help?

Answer 1

We can rewrite this as $\sum \limits_{k=1}^\infty ( \frac{1}{p_k^2}+\frac{1}{p_k^3}+\dots)$.

So we want $\sum\limits_{k=2}^\infty P(k)$ where $P$ is the prime zeta function.