Values of zeta at odd positive integers and Borel's computations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:11:09Z http://mathoverflow.net/feeds/question/38190 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38190/values-of-zeta-at-odd-positive-integers-and-borels-computations Values of zeta at odd positive integers and Borel's computations YBL 2010-09-09T16:11:46Z 2011-01-22T01:57:20Z <p>Someone recently quoted to me this recent <a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an%3Apre05770900&amp;format=complete" rel="nofollow">article</a> that claims to prove that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$. </p> <p>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")</p> <p>Am I mistaken? </p> <p>PS: corrected a typo thx to Pete L Clark </p> http://mathoverflow.net/questions/38190/values-of-zeta-at-odd-positive-integers-and-borels-computations/41876#41876 Answer by Wadim Zudilin for Values of zeta at odd positive integers and Borel's computations Wadim Zudilin 2010-10-12T06:03:51Z 2010-10-12T06:03:51Z <p>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.</p>