Algebraic Relations Between Harmonic Sums and Associated Quantities

We derive the algebraic relations of alternating and non-alternating finite harmonic sums up to the sums of depth~6. All relations for the sums up to weight~6 are given in explicit form. These relations depend on the structure of the index sets of the harmonic sums only, but not on their value. They are therefore valid for all other mathematical objects which obey the same multiplication relation or can be obtained as a special case thereof, as the harmonic polylogarithms. We verify that the number of independent elements for a given index set can be determined by counting the Lyndon words which are associated to this set. The algebraic relations between the finite harmonic sums can be used to reduce the high complexity of the expressions for the Mellin moments of the Wilson coefficients and splitting functions significantly for massless field theories as QED and QCD up to three loop and higher orders in the coupling constant and are also of importance for processes depending on more scales. The ratio of the number of independent sums thus obtained to the number of all sums for a given index set is found to be $\leq 1/d$ with $d$ the depth of the sum independently of the weight. The corresponding counting relations are given in analytic form for all classes of harmonic sums to arbitrary depth and are tabulated up to depth $d=10$.Comment: 39 pages LATEX, 1 style fil

The multiple zeta value data mine

We provide a data mine of proven results for Multiple Zeta Values (MZVs) of the form ζ (s1, s2, ..., sk) = ∑n1 > n2 > ⋯ > nk > 0∞ {1 / (n1s1... nksk)} with weight w = ∑i = 1k si and depth k and for Euler sums of the form ∑n1 > n2 > ... > nk > 0∞ {(ε{lunate}1n1... ε{lunate}1nk) / (n1s1... nksk)} with signs ε{lunate}i = ± 1. Notably, we achieve explicit proven reductions of all MZVs with weights w ≤ 22, and all Euler sums with weights w ≤ 12, to bases whose dimensions, bigraded by weight and depth, have sizes in precise agreement with the Broadhurst-Kreimer and Broadhurst conjectures. Moreover, we lend further support to these conjectures by studying even greater weights (w ≤ 30), using modular arithmetic. To obtain these results we derive a new type of relation for Euler sums, the Generalized Doubling Relations. We elucidate the "pushdown" mechanism, whereby the ornate enumeration of primitive MZVs, by weight and depth, is reconciled with the far simpler enumeration of primitive Euler sums. There is some evidence that this pushdown mechanism finds its origin in doubling relations. We hope that our data mine, obtained by exploiting the unique power of the computer algebra language form, will enable the study of many more such consequences of the double-shuffle algebra of MZVs, and their Euler cousins, which are already the subject of keen interest, to practitioners of Quantum Field Theory, and to mathematicians alike