Multiple zeta values: an introduction (Q5939728)

For positive integers \(s_1,\ldots,s_k\) with \(s_1\geq 2\), the multiple zeta values are NEWLINE\[NEWLINE\zeta(s_1,\ldots,s_k)=\sum_{n_1>\ldots>n_k\geq 1} n_1^{-s_1}\cdots n_k^{-s_k}.NEWLINE\]NEWLINE The product of two zeta values is a sum of zeta values which is obtained by multiplying out the two series (e.g. \(\zeta(s)\zeta(s')=\zeta(s,s')+\zeta(s',s)+\zeta(s+s')\)). Another linear relation arises from the iterated integral representation for \(\zeta({\mathbf s})\). Write \(x_{\mathbf s}=x_{\varepsilon_1}\cdots x_{\varepsilon_p}\) where \({\mathbf s}=(s_1,\ldots,s_k)\), \(p=s_1+\cdots+s_k\) and \(\varepsilon=0\) or 1. Set \(\omega_0(t)=dt/t\) and \(\omega_1(t)=dt/(1-t)\) and let \(\Delta_p\) denote the simplex \(\{{\mathbf t}:1>t_1>\cdots>t_p>0\}\) in \(\mathbb{R}^p\). Then NEWLINE\[NEWLINE\zeta({\mathbf s})=\int_{\Delta_p}\omega_{\varepsilon_1}(t_1)\cdots \omega_{\varepsilon_p}(t_p).NEWLINE\]NEWLINE Writing this as an iterated integral leads to linear relations such as \(\zeta(2)\zeta(3)=\zeta(2,3)+3\zeta(3,2)+6\zeta(4,1)\). The main conjecture is that these two types of relations are sufficient to describe all algebraic relations between these numbers. The paper goes on to explore some algebraic implications of these ideas. There are intriguing connections with the theories of polylogarithms of \textit{A. B. Goncharov} [Math. Res. Lett. 5, 497-516 (1998; Zbl 0961.11040)] and \textit{D. B. Zagier} [Proc. First European Congress of Mathematicians, Vol. 2, Prog. Math. 120, 497-512 (1994; Zbl 0822.11001)].