Is $\sum_\limits{p}^{\infty}\frac{1}{p^{2}}$ irrational where p is prime? How to prove it?
2026-03-26 22:14:57.1774563297
Is $\sum_{p}\frac{1}{p^{2}}$ irrational?
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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.