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

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: ${\sum }_{n}E(k+2n,k)x^n={\sum }_{d|k}\mu (d)(1-x^d{)}^{-k/d}/k$, where $\mu$ 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 $S(l,k)={\sum }_{n<k}\left(\genfrac{}{}{0ex}{}{\lfloor (l+n-1)/2\rfloor }{n}\right)$. Analytical methods, using Tony Hearn's REDUCE, achieve this reduction for the 3698 convergent double Euler sums with $l\le 44$; numerical methods, using David Bailey's MPPSLQ, achieve it for the 1457 convergent $k$-fold sums with $l\le 7$; combined methods yield bases for all remaining search spaces with $S(l,k)\le 34$. 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 $l\le 7$ 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.

Comments: 34 pages, LaTeX