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](http://doi.org/10.12732/ijpam.v91i1.11), [zbMATH entry](https://zbmath.org/1286.11008).

See this [question][2].


  [1]: http://doi.org/10.12732/ijpam.v91i1.11
  [2]: https://mathoverflow.net/questions/38190/values-of-zeta-at-odd-positive-integers-and-borels-computations