TITLE: Is $\sum_{p}\frac{1}{p^{2}}$ irrational?
QUESTION [6 upvotes]: Is $\sum_\limits{p}^{\infty}\frac{1}{p^{2}}$ irrational where p is prime? How to prove it?

REPLY [5 votes]: The function $P(s)=\sum_p \frac{1}{p^s}$ is called the prime zeta function. Unfortunately the value $P(2)$ does not have a known value like $\zeta(2)=\frac{\pi^2}{6}$. Similarly for $P(3), P(4)$ etc. For a related discussion see this MO-question. I think it is not known whether or not $P(2)$ is irrational.