Sum of a certain series related to the primes

Question

It is well known that $$\sum_{n > 0}\frac{1}{n}$$ diverges, but $$\sum_{n > 0}\frac{1}{n^2} = \frac{\pi^2}{6}$$ converges. Similarly, $$\sum_{p}\frac{1}{p}$$ diverges, but $$\sum_{p} \frac{1}{p^2}$$ clearly converges. Is any simple closed form known for this sum, like the one for $\zeta(2)$?

Answer 1

There is the Prime Zeta Function for this:

$$P(2) = 0.4522474220041065...$$

In fact (according to the link),

$$P(2)=\sum_{k=1}^{\infty}{\mu(k)\over k}\log(\zeta(2k))=0.4522474200\dots$$

where $\mu(k)$ is the Mobius Function. In general,

$$P(s)=\sum_{k=1}^{\infty}{\mu(k)\over k}\log(\zeta(ks))$$