Lyndon words, polylogarithms and the Riemann ζ function

AbstractThe algebra of polylogarithms (iterated integrals over two differential forms ω0=dz/z and ω1=dz/(1−z)) is isomorphic to the shuffle algebra of polynomials on non-commutative variables x0 and x1. The multiple zeta values (MZVs) are obtained by evaluating the polylogarithms at z=1. From a second shuffle product, we compute a Gröbner basis of the kernel of this evaluation morphism. The completeness of this Gröbner basis up to order 12 is equivalent to the classical conjecture about MZVs. We also show that certain known relations on MZVs hold for polylogarithms