Multiple zeta function

In mathematics, multiple zeta functions are (English: multiple zeta functions) a generalization of the Riemann zeta - function defined by

The above series converges if Re ( s1) ... Re ( si) > i for all i, it can be defined (analogous to the Riemann zeta function) by analytic continuation as a meromorphic function.

The values ​​for positive integers s1, ..., sk with s1 > 1 are as multiple zeta values ​​(English: multiple zeta values ​​, MZVs ) refers. Called n = S1 ... sk, the "weight" and k the "length" of the argument.

The multiple zeta functions were first defined in the correspondence between Leonhard Euler and Christian Goldbach. Euler proved the reduction formula for:

For example,.

Generally, one can, if is odd, which represent two- zeta function as a rational linear combination of and.

A conjecture of Alexander Goncharov said that the periods of leave of more than unbranched mixed Tate motives be represented as linear combinations of values ​​of multiple zeta function. For the special case of the range defined by the moduli space of curves of genus 0 with marked points and the relative cohomology Tate motif this was first proved in his dissertation in 2007 by Francis Brown. The general form of Goncharovs conjecture proved Brown then in a work published in 2012 in Annals of Mathematics.