In this [article][1], Takaaki Musha proves that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$. I haven't read it so I can't say more.

[Edit: published reference: Musha, Takaaki.
*Negation of the conjecture for odd zeta values*. Int. J. Pure Appl. Math. 91, No. 1, 103-111 (2014). Not referenced by MathSciNet, and referenced by zbMATH ([link](https://zbmath.org/1286.11008)).]

See this [question][2].


  [1]: http://www.ijpam.eu/contents/2014-91-1/11/
  [2]: https://mathoverflow.net/questions/38190/values-of-zeta-at-odd-positive-integers-and-borels-computations