Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k

TL;DR

The paper addresses the evaluation of $k$-fold Euler/Zagier sums with sign variations, which arise in knot theory and quantum field theory. It adopts a data-driven approach, compiling high-precision numerics and deriving generating-function frameworks to organize known and conjectured identities, as well as providing proofs sketches for several results. Key contributions include explicit arbitrary-depth evaluations, new two-parameter self-dual families, and extensive depth-specific classifications and reducibility conjectures, all supported by numerical evidence and appendix sketches. This work advances understanding of the algebraic structure of multidimensional zeta/harmonic sums and provides a foundation for proving further results and guiding applications in mathematics and theoretical physics.

Abstract

Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble results for Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of arbitrary depth, including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations, we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking, we have sketched derivations of all results that have so far been proved.