Values of zeta at odd positive integers and Borel's computations

2010-09-09T16:11:46Z

Someone recently quoted to me this recent article that claims to prove that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$.

I always assumed this was well known. More precisely I thought this result followed from the fact that the regulator $$ K_{2n-1}(\mathbb{Z})\otimes \mathbb{Q} = Ext^1_{MT(\mathbb{Z})}(\mathbb{Q}(0),\mathbb{Q}(n)) \longrightarrow Ext^1_{MHS}(\mathbb{Q}(0),\mathbb{Q}(n)) = \mathbb{C}/(2\pi i)^n\mathbb{Q} $$ is injective (this is usually quoted as a consequence of Borel's computations of K-groups "Stable real cohomology of arithmetic groups", "Cohomologie de $SL_n$ et valeurs de fonctions zêta aux points entiers")

Am I mistaken?

PS: corrected a typo thx to Pete L Clark

Answer by Wadim Zudilin for Values of zeta at odd positive integers and Borel's computations

2010-10-12T06:03:51Z

It is not known (but conjectured) whether the numbers $\zeta(2n+1)/\pi^{2n+1}$ are irrational, $n=1,2,\dots$. It is not even known whether at least one of these numbers is irrational! In fact, the most general (folklore) conjecture states that $\pi$ and all odd zeta values are algebraically independent over $\mathbb Q$. There are natural links between this conjecture and the expected structure of the so-called multiple zeta values; the references I have in mind are papers by A. Goncharov and surveys/talks by M. Waldschmidt.

Values of zeta at odd positive integers and Borel's computations - MathOverflow

Values of zeta at odd positive integers and Borel's computations

Answer by Wadim Zudilin for Values of zeta at odd positive integers and Borel's computations