Special Values of Multiple Polylogarithms

TL;DR

This paper develops a unifying framework for multiple polylogarithms, Euler sums, and related functions, enabling the proof of several conjectured evaluations and deepening understanding of their interrelations.

Contribution

It introduces a general theoretical framework that connects various special values of multiple polylogarithms and proves several longstanding conjectures.

Findings

Proved several previously conjectured evaluations.

Unified understanding of polylogarithms, Euler sums, and zeta functions.

Validated a conjecture of Don Zagier.

Abstract

Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.