Analytic continuation of multiple polylogarithms in positive characteristic

Analytic continuation of multiple polylogarithms

It is not hard to see that this function can be analytically continued to a multi-valued memomorhpic function on C. When n > 1 the special value Lin(1) is nothing else but the Riemann zeta value ζ(n), which has great importance in number theory. In recent years, there has been revival of interest in multi-valued classical polylogarithms and their single-valued cousins. For any positive integers...

5 F eb 2 00 3 Analytic continuation of multiple polylogarithms ∗

In this paper we shall define the analytic continuation of the multiple polylogarithms by using Chen’s theory of iterated path integrals and compute the monodromy of all multiple logarithms explicitly.

Analytic continuation of multiple Hurwitz zeta functions

We use a variant of a method of Goncharov, Kontsevich, and Zhao [Go2, Z] to meromorphically continue the multiple Hurwitz zeta function ζd(s; θ) = ∑ 0<n1<···<nd (n1 + θ1) −s1 · · · (nd + θd)d , θk ∈ [0, 1), to C, to locate the hyperplanes containing its possible poles, and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of ζd(s; θ).

Analytic Continuation of Multiple Zeta Functions

In this paper we shall define the analytic continuation of the multiple (Euler-Riemann-Zagier) zeta functions of depth d: ζ(s1, . . . , sd) := ∑ 0 1 and ∑d j=1 Re (sj) > d. We shall also study their behavior near the poles and pose some open problems concerning their zeros and functional equations at the end.

Analytic Continuation for Cubic Multiple Dirichlet Series