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.

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