In the below article Takaaki Musha claims a proof that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$. I haven't read it so I can't say more.
- Musha, Takaaki. Negation of the conjecture for odd zeta values. Int. J. Pure Appl. Math. 91, No. 1, 103-111 (2014) doi:10.12732/ijpam.v91i1.11, zbMATH entry.
See this question.
