On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory

TL;DR

This work derives a remarkably simple generating-function formula for the number of irreducible k-fold Euler sums across all sign patterns and levels, and a closed-form expression for the search-space size, linking combinatorics to knot theory via a knot/field-theory correspondence. It combines extensive high-precision numerics (PSLQ-based) with analytical tools in a two-pronged approach (anterior numerics and posterior analytics) to enumerate irreducibles, construct concrete bases, and establish rigorous bounds up to substantial levels. The results illuminate the role of Euler sums in perturbative quantum electrodynamics, support Dirk Kreimer's knot–field theory program, and reveal deep connections to knot theory and number theory, including explicit representations of certain sums in terms of zeta-values and knot-numbers. The work provides practical bases for reducing Euler sums in multi-loop calculations and advances our understanding of the transcendental content of field-theoretic counterterms up to 13 loops.

Abstract

A generating function is given for the number, $E(l,k)$, of irreducible $k$-fold Euler sums, with all possible alternations of sign, and exponents summing to $l$. Its form is remarkably simple: $\sum_n E(k+2n,k) x^n = \sum_{d|k}μ(d) (1-x^d)^{-k/d}/k$, where $μ$ is the Möbius function. Equivalently, the size of the search space in which $k$-fold Euler sums of level $l$ are reducible to rational linear combinations of irreducible basis terms is $S(l,k) = \sum_{n<k}{\lfloor(l+n-1)/2\rfloor\choose n}$. Analytical methods, using Tony Hearn's REDUCE, achieve this reduction for the 3698 convergent double Euler sums with $l\leq44$; numerical methods, using David Bailey's MPPSLQ, achieve it for the 1457 convergent $k$-fold sums with $l\leq7$; combined methods yield bases for all remaining search spaces with $S(l,k)\leq34$. These findings confirm expectations based on Dirk Kreimer's connection of knot theory with quantum field theory. The occurrence in perturbative quantum electrodynamics of all 12 irreducible Euler sums with $l\leq 7$ is demonstrated. It is suggested that no further transcendental occurs in the four-loop contributions to the electron's magnetic moment. Irreducible Euler sums are found to occur in explicit analytical results, for counterterms with up to 13 loops, yielding transcendental knot-numbers, up to 23 crossings.