The multiple zeta value data mine

Blümlein, J.; Broadhurst, D. J. and Vermaseren, J. A. M. (2010). The multiple zeta value data mine. Computer Physics Communications, 181(3) pp. 582–625.

DOI: https://doi.org/10.1016/j.cpc.2009.11.007

Abstract

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 / (n₁^s1... n_k^sk)} with weight w = ∑_{i = 1}^k s_i and depth k and for Euler sums of the form ∑_{n1 > n2 > ... > nk > 0}^∞ {(ε{lunate}₁^n1... ε{lunate}₁^nk) / (n₁^s1... n_k^sk)} 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.

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About

• Item ORO ID
• 19392
• Item Type
• Journal Item
• ISSN
• 0010-4655
• Project Funding Details

• Funded Project Name	Project ID	Funding Body
  Not Set	Not Set	Dutch Foundation for Fundamental Research of Matter (FOM)
  Not Set	Not Set	DFG Sonderforschungsbereich Transregio 9
  Not Set	Not Set	Computergestützte Theoretische Teilchenphysik
  Not Set	Not Set	European Commission MRTN HEPTOOLS [contract number MRTN-CT-2006-035505]

• Academic Unit or School
• Faculty of Science, Technology, Engineering and Mathematics (STEM) > Physical Sciences
  Faculty of Science, Technology, Engineering and Mathematics (STEM)
• Copyright Holders
• © 2009 Crown Copyright
• Depositing User
• David Broadhurst