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Oct 16, 2015 at 19:15 comment added GH from MO @NoamD.Elkies: Yes, I meant what is proven. Probably the odd zeta values are even algebraically independent, which shows in particular that the OP's question is too broad (in addition to being too vague).
Oct 16, 2015 at 18:57 comment added Noam D. Elkies "Unknown" meaning unproven, yes, but it's morally certain that $\zeta(5) \notin \bf Q$...
Oct 16, 2015 at 18:36 history answered GH from MO CC BY-SA 3.0