The Multiple Zeta Value data mine

By J. Blumlein, David John Broadhurst, Jos Vermaseren. Computer Physics Communications, 2009

363 citations recorded by the source corpus.

Abstract We provide a data mine of proven results for Multiple Zeta Values (MZVs) of the form ζ ( s 1 , s 2 , … , s k ) = ∑ n 1 > n 2 > ⋯ > n k > 0 ∞ { 1 / ( n 1 s 1 ⋯ n k s k ) } with weight w = ∑ i = 1 k s i and depth k and for Euler sums of the form ∑ n 1 > n 2 > ⋯ > n k > 0 ∞ { ( ϵ 1 n 1 ⋯ ϵ 1 n k ) / ( n 1 s 1 ⋯ n k s k ) } with signs ϵ 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/8e3bd9d304376062f76a9114b4c1c19c05d24c05

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Citations in the corpus

Motivic multiple zeta values and superstring amplitudes (2012, methodology): “a number of basis elements needs at least two even entries [19].”
Standard relations of multiple polylogarithm values at roots of unity (2007, background): “Let Ax be the algebra of A together with the multiplication defined by shuffle product x. Denote the subalgebra A0 by A0 x when we consider the shuffle product.”
Single-valued harmonic polylogarithms and the multi-Regge limit (2012, background): “In fact, it is known that in the special case N → ∞, in which the sums reduce to multiple zeta values, many new relations emerge [62, 74–76].”
Graphical functions and single-valued multiple polylogarithms (2013, neutral): “Their calculation is quick and up to loop order 12 the MZVs in the result are of weights less than or equal to 21 which allows us to reduce them to a standard basis by the datamine [3] or by zeta procedures [39].”
Iterated Binomial Sums and their Associated Iterated Integrals (2014, background): “appears beyond the multiple zeta values [16] and the special numbers in the case of cyclotomic and generalized harmonic sums [24,26].”
Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals (2014, background): “3For MZV and alternating sums, [38] provides reductions up to weights 22 and 12, respectively.”
Feynman integrals and hyperlogarithms (2015, methodology): “To compute ∫∞ 0 Li1,1(−x/y,−y) y(1+y) dy, one can enter > convert(Mpl([1,1],[-x/y,-y])/y/(y+1), HlogRegInf): > integrationStep(%, y): > fibrationBasis(%, [x]); 1For MZV and alternating sums, [28] provides reductions up to weights 22 and 12, respectively.”
Polylogarithms, multiple zeta values and superstring amplitudes (2013, methodology): “The accompanying articles and the formulae presented in [41, 42] make use of harmonic polylogarithms, which are - up to a sign - multiple polylogarithms with entries ai ∈ {−1, 0, 1}: H(~a; z) = (−1)G(~a; z) ai ∈ {−1, 0, 1} (5.”
Multiple zeta values vs. multiple zeta-star values (2011, background): “(The readers can also find related topics in [3] and [8].”
Mathematical aspects of scattering amplitudes (2014, background): “ can, at least in principle, solve the double-shuﬄe relations for each weight, and in this way we can construct an explicit basis for each Zn. This has been done explicitly up to high weights in ref. [22]. Moreover, there is a conjecture about the dimensions dn = dimQZn: d1 = 0, d0 = d2 = 1, dk = dk”
An odd variant of multiple zeta values (2016, methodology): “Using the Multiple Zeta Value Data Mine [4], we have obtained expressions for multiple t-values through weight 7, which we list in the Appendix below.”