Evaluation of Triple Euler Sums

Abstract

Let $a,b,c$ be positive integers and define the so-called triple, double and single Euler sums by $$\zeta(a,b,c) \ := \ \sum_{x=1}^{\infty} \sum_{y=1}^{x-1} \sum_{z=1}^{y-1} {1 \over x^a y^b z^c},$$ $$ \zeta(a,b) \ := \ \sum_{x=1}^\infty \sum_{y=1}^{x-1} {1 \over x^a y^b} \quad $$ and $$ \zeta(a) \ := \ \sum_{x=1}^\infty {1 \over x^a}.$$ Extending earlier work about double sums, we prove that whenever $a+b+c$ is even or less than 10, then $\zeta(a,b,c)$ can be expressed as a rational linear combination of products of double and single Euler sums. The proof involves finding and solving linear equations which relate the different types of sums to each other. We also sketch some applications of these results in Theoretical Physics.

Compact analytical form for non-zeta terms in critical exponents at order 1/N3

“…This is sobering news for colleagues seeking to push back the computational frontier in the perturbative sector [38] of Chern-Simons theory. At present this lags far behind the 7-loop [23] level, achieved for φ 4 -theory with D = 4, in an analysis that spectacularly confirmed Kreimer's predictions [17,18] for the fascinating nexus of knot/number/field theory [20], whose study has advanced with great rapidity [4,7,9,10,11,12,19,22,26,27,35,36] in recent months.…”

“…The even-dimensional ε-expansions of BGK [4] were confirmed up to weight 19, making the probability of error in the new result (3), or in the BGK result, negligible. Checking their equivalence had a beneficial spin-off for number theory [9]. It is remarkable how closely the massless three-dimensional results mimic massive four-dimensional results [33,34].…”

“…The four possible choices of sign were not treated on an equal footing in [8], where the reduction of alternating sums was left in a 'somewhat more implicit state', compared with the nonalternating case. The reader is referred to a recent compendium [9] of results for Euler sums, where this field-theory byproduct appears as Equation (74), in a notation which unifies all 8 cases that result from the four possible choices of sign, and the two possible choices of the opposite parities of the exponents in odd-weight double sums.…”

“…The situation is thoroughly reviewed by Broadhurst, Gracey and Kreimer in [4] (hereafter BGK), who first showed that the ε-expansions of I(1 − ε) and I(2 − ε), for the σ-model and φ 4 -theory, entail double Euler sums [5], of the type ζ(s, t) = m>n>0 m −s n −t . The majority of such sums are conjectured [6,7,8,9,10,11,12] to be irreducible to the single sums ζ(r) = n>0 n −r , when s and t have the same parity, the weight s + t exceeds 6, and s > t > 1. Prior to BGK, the first double-sum irreducible, conjectured [13] to be at weight 8, had not been detected by large-N studies [14,15,16].…”

Explicit formulas of Euler sums via multiple zeta values

“…Setting m = 2 in Theorem 2.1, and noting that C 2 = {[2], [1,1]} and S 2 = {(1)(2), (1, 2)}, we obtain (3.9). According to Borwein and Girgensohn [10], the MZVs ζ(q, i 1 , i 2 ) and ζ(q, i 2 , i 1 ) are reducible to zeta values and MZVs of depth 2 when the weight w = i 1 + i 2 + q is even, then for i 1 , i 2 > 1, the sums S i 1 i 2 ,q are reducible to linear sums and zeta values. This is in fact one of the Flajolet-Salvy theorems [21,Theorem 4.2].…”