The inversion formula of polylogarithms and the Riemann-Hilbert problem

TL;DR

This paper develops a recursive Riemann-Hilbert approach to reconstruct polylogarithms from zeta values and interprets the Knizhnik-Zamolodchikov connection problem within this framework, offering new insights into their interrelations.

Contribution

It introduces a novel recursive Riemann-Hilbert method for polylogarithm reconstruction and links the KZ equation connection problem to this approach.

Findings

Reconstruction of polylogarithms from zeta values using Riemann-Hilbert problem

Framework for interpreting KZ connection problem as a Riemann-Hilbert problem

Establishment of a recursive additive-type Riemann-Hilbert method

Abstract

In this article, we set up a method of reconstructing to polylogarithms $\mathrm{Li}_k(z)$ from zeta values $\zeta(k)$ via the Riemann-Hilbert problem. This is referred to as "a recursive Riemann-Hilbert problem of additive type." Moreover, we suggest a framework of interpreting the connection problem of the Knizhnik-Zamolodochikov equation of one variable as a Riemann-Hilbert problem.