Multiple zeta function

In the mathematics are multiple zeta functions (engl .: multiple functions zeta ) is a generalization of the Riemann zeta function is defined by

${\displaystyle \zeta (s_{1},\ldots ,s_{k})=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ {\frac {1}{n_{1}^{s_{1}}\cdots n_{k}^{s_{k}}}}=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ \prod _{i=1}^{k}{\frac {1}{n_{i}^{s_{i}}}},\!}$

The above series converges if for all , it can be defined (analogously to the Riemann zeta function) by analytical continuation as a meromorphic function on . ${\displaystyle Re(s_{1})+\ldots +Re(s_{i})>i}$ ${\displaystyle i}$ ${\displaystyle \mathbb {C} ^{n}}$

The values for positive integer with are called Multiple Zeta Values (ger .: multiple zeta values referred MZVs). One calls the "weight" and the "length" of the argument. ${\displaystyle s_{1},\ldots ,s_{k}}$ ${\displaystyle s_{1}>1}$ ${\displaystyle n=s_{1}+\ldots +s_{k}}$ ${\displaystyle k}$

The multiple zeta functions were first defined in the correspondence between Leonhard Euler and Christian Goldbach . Euler proved the reduction formula for : ${\displaystyle 1<s\in \mathbb {Z} }$

${\displaystyle \zeta (s,1)={\frac {1}{2}}s\zeta (s+1)+{\frac {1}{2}}\sum _{k=2}^{s-1}\zeta (k)\zeta (s+1-k)}$.

For example is . ${\displaystyle \zeta (2,1)=\zeta (3)}$

In general, if is odd, the twofold zeta function can be represented as a rational linear combination of and with . ${\displaystyle m+n}$ ${\displaystyle \zeta (m,n)}$ ${\displaystyle \zeta (m+n)}$ ${\displaystyle \zeta (k)\zeta (m+n-k)}$ ${\displaystyle k\in \mathbb {N} }$

A conjecture by Alexander Goncharov said that the periods of over unbranched mixed Tate motifs can be represented as linear combinations of values ​​of the multiple zeta function. For the special case of the Tate motif defined by the module space of curves of gender 0 with marked points and the relative cohomology , this was first proven by Francis Brown in his dissertation in 2007. Brown then proved the general form of Goncharov's conjecture in a paper published in Annals of Mathematics in 2012 . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Q} \left[{\frac {1}{2\pi i}}\right]}$ ${\displaystyle {\mathcal {M}}_{0,n}}$ ${\displaystyle n}$ ${\displaystyle H^{l}({\overline {\mathcal {M}}}_{0,n}-A,B-B\cap A)}$

literature

1. ^ Goncharov: Multiple polylogarithms and mixed Tate motives
2. ↑ Brown: Multiple zeta values ​​and periods of moduli spaces , Annales Scientifiques de l'ENS, Volume 42, 2009, pp. 371-489, abstract
3. ↑ Brown: Mixed Tate motives over Z

Web links

Deligne: "Le groupe fondamental de la droite projective moins trois points" (PDF; 4.4 MB) explains the connection between mixed tate motifs and multiple zeta functions.