Series of the form $\sum \frac {1}{{p_k}^n}$

Question

Is it reasonable to expect that for every $n \in \mathbb N \setminus \{1\}$ the series $\sum \dfrac {1}{{(p_k)}^n}$ converges to an irrational number? Here, $p_k$ is $k$-th prime.

Are there any results in this direction?

Answer 1

The function you mention in your question is the Prime Zeta function, defined as

$$ P(n)=\sum_{k>1} p_k^{-n} $$

for $Re(n)>1$.

The irrationality of its values is still an open problem.