The Lerch zeta function III. Polylogarithms and special values

TL;DR

This paper explores the algebraic and analytic properties of the Lerch zeta function and its relation to polylogarithms, including its analytic continuation, monodromy, and deformations for special integer parameters.

Contribution

It extends the complex variable analysis of the Lerch zeta function, computes its monodromy, and introduces a deformation of polylogarithms with a detailed monodromy representation.

Findings

Analytic continuation of the Lerch transcendent to a maximal domain

Explicit computation of monodromy functions for the multivalued Lerch zeta function

Identification of a non-algebraic deformation of polylogarithm monodromy

Abstract

This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $\Phi(s, z, c)$ is obtained from the Lerch zeta function $\zeta(s, a, c)$ by the change of variable $z=e^{2 \pi i a}$. We show that it analytically continues to a maximal domain of holomorphy in three complex variables $(s, z, c)$ as a multivalued function defined over the base manifold ${\bf C} \times P^1({\bf C} \smallsetminus \{0, 1, \infty\}) \times ({\bf C}\smallsetminus {\bf Z})$. and compute the monodromy functions defining the multivaluedness. For positive integer values s=m and c=1 this function is closely related to the classical m-th order polylogarithm $Li_m(z)$ We study its behavior as a function of two variables $(z, c)$ for special values where s=m is an integer. For $m \ge 1$ it gives a one-parameter deformation of the polylogarithm, and satisfies a linear ODE with coefficients depending on c, of order m+1 of Fuchsian type. We determine its (m+1)-dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of $Li_m(z).$