Closed form for $\sum_1^\infty 1/p^n$ I was wondering if there are some studies on closed forms for the sum 
 $$\sum_{p \in \mathbb{P}}^\infty \frac{1}{p^n},$$
where $\mathbb{P}$ denotes the set of prime numbers. 
Obviously I know that for $n=1$ the series is divergent, but I have found out that it converges for $n>1$; hence the question if there any known closed forms to express the sum.

There is a trivial closed form (it follows from the Euler's product identity) for
$$\sum_{p\in\mathcal{P}}-\log\left(1-\frac{1}{p^n}\right) = \log\zeta(n) $$
and:
$$\sum_{p\in\mathcal{P}}\log\left(1+\frac{1}{p^n}\right) = \log\zeta(n)-\log\zeta(2n) $$
hence a pretty good approximation (it is an upper bound) for the original series is given by:
$$\sum_{p\in\mathcal{P}}\frac{1}{p^n}\approx \log\zeta(n)-\frac{1}{2}\log\zeta(2n)$$
and an exact identity can be derived by expressing $z$ as a linear combination of $\log(1\pm z^k)$ with $k\in\mathbb{N}^*$.
