The Multiple Zeta Value Data Mine

Blumlein, J.; Broadhurst, D.J.; Vermaseren, J.A.M.

doi:10.1016/j.cpc.2009.11.007

The Multiple Zeta Value Data Mine

J. Blumlein
(
- DESY, Zeuthen
)
,
D.J. Broadhurst
(
- Open U., England
)
,
J.A.M. Vermaseren
(
- DESY, Zeuthen and
- NIKHEF, Amsterdam
)

Jul, 2009

77 pages

Published in:

Comput.Phys.Commun. 181 (2010) 582-625

e-Print:

0907.2557 [math-ph]

DOI:

10.1016/j.cpc.2009.11.007

Report number:

DESY-09-003,
SFB-CPP-09-65

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Abstract: (arXiv)

We provide a data mine of proven results for multiple zeta values (MZVs) of the form

\zeta(s_1,s_2,...,s_k)=\sum_{n_1>n_2>...>n_k>0}^\infty \{1/(n_1^{s_1} >... n_k^{s_k})\}

with weight

w=\sum_{i=1}^k s_i

and depth

k

and for Euler sums of the form

\sum_{n_1>n_2>...>n_k>0}^\infty t\{(\epsilon_1^{n_1} >...\epsilon_1 ^{n_k})/ (n_1^{s_1} ... n_k^{s_k}) \}

with signs

\epsilon_i=\pm1

. Notably, we achieve explicit proven reductions of all MZVs with weights

w\le22

, and all Euler sums with weights

w\le12

, 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\le30

), 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 {\sc 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.

zeta function
analytic properties
duality
master equation
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