Nested Series and Iterated Integrals

Abstract. Recently, several extensions of polylogarithms and the Riemann zeta function in the direction of multiple harmonic series have been studied. These functions satisfy many remarkable identities which, in a sense, arise because the underlying functions can be represented both as nested series and as iterated integrals. Nevertheless, there are several beautiful conjectures that remain to be proved; and we will describe a few of them here. We also indicate how Don Zagier's longstanding conjecture, which can be viewed as a multidimensional, non-commutative generalization of Euler's formula for $\zeta(2n)$, was finally resolved.