Multiple polylogarithms, cyclotomy and modular complexes

The multiple ζ-values were invented and studied by Euler [E] and then forgotten. They showed up again in such different subjects as quantum groups [Dr] (the Drinfeld associator), Zagier’s studies [Z1-2], the Kontsevich integrals for Vassiliev knot invariants, mixed Tate motives over Spec Z [G1-2], and, recently, in computations in quantum field theory [B], [Kr]. The multiple polylogarithms were studied in [G1-4]. In this paper we investigate them at N -th roots of unity: x1 = . . . = x N m = 1. Notice that Li1(x) = − log(1 − x), so if ζN is a primitive N -th root of 1, then Li1(ζN ) is a logarithm of a cyclotomic unit in Z[ζN , N−1]. In general the supply of numbers we get coincides with the linear combinations of multiple Dirichlet L-values