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

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

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

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

mightbe 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? $\endgroup$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". $\endgroup$5more comments