# Is the number $\sum_{p\text{ prime}}p^{-2}$ known to be irrational?

Is the number $$\sum_{p\text{ prime}}p^{-2}$$ known to be irrational?

The limit exists, since $$\sum_{p\text{ prime}}p^{-2}<\sum_{i=1}^{\infty}i^{-2}=\frac{\pi^{2}}{6}$$.

• relevant answer here: mathoverflow.net/questions/53443/… Jun 15, 2016 at 15:58
• No it does not, but that's because a fool can ask more questions than 1000 wise men can answer. Jun 15, 2016 at 16:10
• I would consider this question as part of transcendental number theory (as one really expects this number to be transcendental). The main takeaway from Franz's last comment is that in fact very little is known in this area: we don't even know whether $e - \pi$ is irrational. But the question might be answered conditionally upon assuming some standard conjecture in the area (e.g. Schanuel's conjecture), just as one sometimes proves things conditional on the truth of the Riemann hypothesis if no unconditional proof is known. Is this question important to you? Jun 15, 2016 at 16:20
• If it's important to you, then could you please give some details why? Giving some context can only improve this question, and indicate why the question is not just one of idle curiosity (that any "fool" could ask, harking back again to Franz's comment). Jun 15, 2016 at 16:31
• Here might be my own question on this: is this number known to be a period (in the sense used by Kontsevich and Zagier)? I'm going to quote from their paper ihes.fr/~maxim/TEXTS/Periods.pdf (page 16): "... many, if not almost all proofs of irrationality and transcendence results use periods and their associated differential equations in one form or another". Jun 15, 2016 at 16:53