What is already known for $\zeta(n)$, $n\in 2\mathbb{N}+1$

Question

Apéry showed $\zeta(3)\notin\mathbb{Q}$. What is also known or conjectured for $\zeta(n)$ with n odd? Is for example something known for $\zeta(5)$? Is there a theorem that says 'at least one of $\zeta(n),\zeta(m),...\zeta(l)$ is irrational?

Answer 1

This paper will be helpful:

V.V.Zudilin, "An Elementary Proof of Apery's Theorem" (http://arxiv.org/pdf/math/0202159.pdf)

Introduction of this gives various answers for your question.

Answer 2

Yes, Wadim Zudilin proved in 2001 that one of the numbers $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$ is irrational, here the link to the paper