Multiple Polylogarithms: An Introduction

Abstract

Multiple polylogarithms in a single variable are defined by $$L{i_{\left( {{s_1}, \cdots ,{s_k}} \right)}}\left( z \right) = \sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{{{z^{{n_1}}}}} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$ , when s1, … , s k are positive integers and z a complex number in the unit disk. For k = 1, this is the classical polylogarithm Li s (z). These multiple polylogarithms can be defined also in terms of iterated Chen integrals and satisfy shuffle relations. Multiple polylogarithms in several variables are defined for s i ≥ 1 and |z i | < 1(1 ≤ i ≤ k) by $$L{i_{\left( {{s_1}, \cdots ,{s_k}} \right)}}\left( {{z_1}, \cdots {z_k}} \right) = \sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{{z_1^{{n_1}} \cdots z_k^{{n_k}}}} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$ , and they satisfy not only shuffle relations, but also stuffle relations. When one specializes the stuffle relations in one variable at z = 1 and the stuffle relations in several variables at z1 = ⋯ = z k = 1, one gets linear or quadratic dependence relations between the Multiple Zeta Values $$\zeta \left( {{s_1}, \cdots ,{s_k}} \right)\sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{1} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$ which are defined for k, s1, … ,s k positive integers with s1 ≥ 2. The Main Diophantine Conjecture states that one obtains in this way all algebraic relations between these MZV.