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

TL;DR

This paper compiles and analyzes extensive results on Euler/Zagier sums of arbitrary depth, highlighting new empirical findings and providing a foundation for future mathematical and physical research.

Contribution

It presents a comprehensive collection of results for Euler/Zagier sums of any depth, including many empirically derived and potentially new findings, with some proofs sketched.

Findings

Results are likely exhaustive for certain classes of sums.

Many results are empirically obtained and previously unknown.

The paper provides derivations for all proved results.

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.