Lyndon words, polylogarithms and the Riemann \(\zeta\) function. (English) Zbl 0959.68144

Summary: The algebra of polylogarithms (iterated integrals over two differential forms \(\omega_0= dz/z\) and \(\omega_1= dz/(1- z))\) is isomorphic to the shuffle algebra of polynomials on non-commutative variables \(x_0\) and \(x_1\). 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.


11M32 Multiple Dirichlet series and zeta functions and multizeta values
11G55 Polylogarithms and relations with \(K\)-theory
68R15 Combinatorics on words
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