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The Multiple Zeta Value Data Mine

J. Blümlein, D. J. Broadhurst, J. A. M. Vermaseren

arXiv:0907.2557v2math-phcs.MShep-phhep-thmath.AGmath.NT

TL;DR

The paper addresses incomplete relations and basis reductions for MZVs and Euler sums, which matter in mathematics and perturbative quantum field theory. It derives Generalized Doubling Relations and uses computer-assisted calculations to obtain explicit reductions and modular-arithmetic evidence for conjectured basis dimensions across substantial weight and depth ranges.

  • Problem

    Shuffle and stuffle relations alone do not yield the conjectured minimal Euler-sum bases, while existing depth-mixing relations are unsuitable for depth-sensitive calculations.

  • Method

    The paper combines individual shuffle and stuffle relations with Generalized Doubling Relations and computer-assisted basis construction and reduction procedures.

  • Results

    Explicit reductions cover all Euler sums through weight 12 and MZVs through weight 22; modular arithmetic checks extend conjecture tests to higher weights and depths.

  • Takeaways & Limitations

    Generalized Doubling Relations reduce the number of undetermined constants to expected values and support the conjecture that the stated relations suffice for minimal Euler-sum bases.

  • Takeaways & Limitations

    Complete runs are unavailable for weights 27 and 28, where the greatest-depth elements are missing and predictions require two-fold extensions.

Abstract

from arXiv · show

We provide a data mine of proven results for multiple zeta values (MZVs) of the form $ζ(s_1,s_2,...,s_k)=\sum_{n_1>n_2>...>n_k>0}^\infty \{1/(n_1^{s_1} >... n_k^{s_k})\}$ with weight $w=\sum_{i=1}^k s_i$ and depth $k$ and for Euler sums of the form $\sum_{n_1>n_2>...>n_k>0}^\infty t\{(ε_1^{n_1} >...ε_1 ^{n_k})/ (n_1^{s_1} ... n_k^{s_k}) \}$ with signs $ε_i=\pm1$. 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.

1 Introduction

The paper studies relations and basis representations for MZVs and Euler sums, motivated by their mathematical and perturbative QFT applications. It proves extensive reductions and uses computer algebra to investigate conjectured basis dimensions.

  • Motivation: MZVs and Euler sums arise in mathematics and perturbative QFT, where higher-order QED and QCD calculations require related multiple harmonic sums.Euler sums are obtained as limits of related multiple sums.
  • Research problem: The number of Euler sums and MZVs grows rapidly, making relations at fixed weight and depth—and relations connecting the two classes—a central question.The paper also considers pushdown as a degree of freedom quantifying relations between MZVs and Euler sums.
  • Research problem: Shuffle and stuffle relations alone do not produce the conjectured minimal bases; doubling is needed from weight 8, and generalized doubling from weight 11.From weight 12 onward, some relations express MZVs at a given depth through Euler sums of lesser depth.
  • Approach: The paper derives generalized doubling relations, develops computer-algebra methods for systematic reductions, and stores the resulting representations in the Multiple Zeta Value Data Mine.The paper also describes its organization around formalism, generalized doubling relations, FORM code, computational runs, and appendices on bases and pushdowns.
  • Results: All Euler sums through weight 12 and MZVs through weight 22 receive explicit analytic basis representations.The conjectured basis size is checked through weight 24 using modular arithmetic, with additional confirmations at higher weights and selected depths.

2 Basic Formalism

The formalism represents MZVs and Euler sums through nested sums and harmonic polylogarithms, which provide shuffle and stuffle structures for deriving reductions. The paper combines these relations while accounting for divergence and depth behavior.

  • Objects and notation: The paper uses finite nested S-sums, Z-sums, and harmonic polylogarithms, whose limits define MZVs and Euler sums.It generally studies colored objects with numerator signs (±1)^k, corresponding to square roots of unity.
  • Scope: Euler sums are too small in general to represent all Feynman diagrams for no-scale scalar processes, requiring extensions at higher orders.The stated limitation also applies to QCD and QED field theories.
  • Objects and notation: Sums use nested-sum notation, while harmonic polylogarithms use iterated-integral notation based on the alphabet 0, ±1.Index transformations relate the two notations, and the number of indices gives the weight.
  • Algebraic relations: Harmonic polylogarithms obey a shuffle algebra, whereas finite sums obey a stuffle, or quasi-shuffle, algebra with additional diagonal terms.Stuffle relations can be arranged so they do not increase depth and may lower it.
  • Additional relations: At infinity or at unity, triangle rules and doubling relations supply additional relations, with correction terms required when divergent sums occur.H-functions at unity can be rewritten as nested sums and therefore obey both shuffle- and stuffle-type structures.
  • Algebraic relations: The shuffle and stuffle products can be equated to form double-shuffle relations containing only terms of the same weight.For these calculations, the programs use shuffle and stuffle relations separately to improve algorithmic optimization.

3 Conjectures on Bases at Fixed Weight and Depth

The paper summarizes conjectured enumerations of primitive Euler sums and MZVs by weight and depth, then compares its data with these predicted basis dimensions. It reports extensive supporting evidence for the conjectures.

  • Conjectured basis dimensions: Broadhurst and Broadhurst–Kreimer conjectures predict the sizes of bases for Euler sums and MZVs, respectively.The paper reviews these conjectures before presenting its data-mining results.
  • Euler sums: Primitive Euler sums are basis elements from which all Euler sums can be represented polynomially, while independent sums cannot be reduced to lower-depth primitives and products.Different basis representations may be chosen even though the numbers of independent and primitive sums are fixed.
  • Euler sums: The conjectured Euler-sum enumeration uses weight and depth filtrations, with ln(2) and ζ2 serving as seeds for the basis.Möbius transformation is used to obtain the numbers of independent Euler sums from binomial-coefficient expansions.
  • MZVs: MZV basis enumeration is more complicated than the initial generating-function guess because the double-shuffle solution does not match it from weight 11 onward.This mismatch motivates seeking new depth-lowering relations.
  • Evidence: The data mine impressively supports the conjectured generating function, including D12,4 = 1 and D12,2 = 1 and the stated even-weight double-sum counts.The correction numerator enforces the weight-12 values, while the denominator gives D2m,2 = ⌊(m−1)/3⌋.

4 Generalized Doubling Relations

Generalized Doubling Relations (GDRs) are introduced because shuffle, stuffle, and doubling relations cease to produce conjecturally minimal Euler-sum bases from weight 11 onward. The paper derives these depth-lowering relations and shows that they restore the expected basis counts across several weights and depths.

  • Motivation: From w = 11 onward, shuffle, stuffle, and doubling relations leave excess Euler-sum variables, requiring new depth-lowering relations.At w = 11 and w = 12, the GDRs supply the missing equation and recover agreement with the conjectured basis size.
  • Derivation: The GDR derivation generalizes partial-fractioning and summation-range doubling across increasing depths, while retaining sign factors and correcting divergent cases.The construction introduces negative indices through parity factors and includes correction terms when divergent sums or altered summation ranges are involved.
  • Depth-2 impact: At weight 6 and depth 2, 203 shuffle and stuffle equations contribute to the final relation, whereas depth-2 GDRs resolve it directly with those relations.The GDRs therefore reproduce the desired formula without requiring the full large collection of contributing equations.
  • Higher-depth tests: At depth 3, applying GDRs reduces the number of undetermined constants to its expected value and yields complete expressions through w = 51.The highest-weight run required about 20 hours of CPU time on a single 2.33 GHz Xeon processor.
  • Conjectural scope: The paper conjectures that shuffle, stuffle, doubling, and GDRs suffice to reduce Euler sums at fixed weight and depth to minimal sets matching the conjectured dimensions.The authors note that relations leaking across at least two depth units make the problem difficult without GDRs.

5 The Computer Program

The program constructs and incrementally solves sparse systems of shuffle, stuffle, and doubling equations, using grouped elimination and carefully ordered substitutions to reduce MZVs and Euler sums.

  • Program structure: The program builds a master expression containing one term for each target sum, then incorporates generated equations and substitutions incrementally.This avoids keeping all equations in memory simultaneously and retains newly acquired relations in the master expression.
  • Program structure: FORM was selected for sparse polynomial representations, avoiding explicit storage and processing of the many zeroes in the systems.The authors describe the problems as having typically many thousands of zeroes for each non-zero element.
  • Elimination strategy: Grouped Gaussian elimination substitutes several variables simultaneously, reducing repeated pattern matching and controlling costly expression normalization.The method first eliminates above and below the diagonal within equation groups before substituting the remaining variables together.
  • Elimination strategy: Equation ordering starts with low depths, applies stuffles before shuffles, and uses duality to omit MZVs beyond half the weight.Shuffles preserve depth, whereas stuffles may preserve or lower it; experimentally, omitting some stuffles would make the program significantly slower.
  • Performance: Shuffles cause the main intermediate expression swell, especially at depths well below half the weight, while complete runs spend most time on stuffles and limited-depth runs on shuffles.The performance behavior is reported for complete and limited-depth runs, with shuffle complexity dominating the latter.
  • Performance: More than 7 · 10^12 terms were generated in one run lasting over 30 days, demonstrating the scale reached by the representation and modular-arithmetic treatment.The authors characterize this total as a new record.

6 The Running of the Programs

The authors report extensive MZV and Euler-sum computations, combining complete rational runs with modular and depth-limited runs to reach higher weights. The resulting outputs and explicit representations are collected in the data mine.

  • Euler sums: w = 12 covers all 236196 Euler sums, expressed in a Lyndon basis of negative odd integers.
  • Computational limits: The authors stopped the complete Euler-sum runs at w = 12 because output size became the major problem; w = 13 was projected at almost 8 Gbytes.
  • Higher-weight runs: Depth-limited runs are faster and permit calculations at greater weights, although higher runs may omit some basis elements.If the conjecture is correct, the runs at w = 25,26 should still provide complete bases.
  • Computational scale: The w = 18,d = 6 run processed more than 7·10^12 terms and was the most costly run reported.
  • Data availability: The outputs of the reported runs were collected in the data mine, alongside processed files intended to make the results more accessible.

7 The Data Mine

The data mine packages datasets, programs, documentation, and multiple file formats for accessing proven MZV and Euler-sum results. Its organization supports both human-readable use and efficient FORM-based retrieval, but its size imposes substantial practical constraints.

  • Notation: MZVs and Euler sums are represented in FORM using functions and compact symbolic notations, with negative indices denoting alternating sums for Euler sums.
  • Internal representation: The programs use integral notation with binary index encoding for MZVs and ternary encoding for Euler sums to speed calculations.
  • Contents: The data mine contains datasets, usage information, links, run logs, table files, binary files, programs, and example programs.
  • Access formats: Tablebases provide fast access to individual elements without loading complete large tables through the compiler.
  • Practical constraint: The combined compressed files exceed 30 Gbytes, making the data mine's size its main practical problem.

8 FORM Aspects

The authors substantially improve FORM and TFORM for manipulating shuffle, stuffle, conversion, parallelization, modular-arithmetic, and long-running calculations. These changes address expression complexity and computational reliability in large MZV and Euler-sum reductions.

  • Notation conversion: New conversion commands make transformations between H and sum notation noticeably faster and easier to read.
  • Shuffle and stuffle operations: Built-in shuffle commands reduce duplicate-term combinatorics, improving a test calculation from 37.38 seconds to 0.01 seconds.The corresponding generated-term count is 2,496,144 for the slower program and 5,163 for the shuffle-command program.
  • Shuffle and stuffle operations: Stuffle products distinguish Z- and S-sum definitions with appended plus and minus notation, while MZVs can use direct H-based operations.
  • Parallelization: Parallelization becomes nearly ideally efficient for sufficiently large equation groups, despite sequential output and residual load-balancing issues.
  • Modular arithmetic: FORM's modular arithmetic required redesigned and extended facilities, supporting large computations with a 31-bit prime modulus.
  • Validation: The authors report that FORM failures were obvious crashes, while repeated runs and nondeterministic worker assignments provided additional checks.
  • Validation: Independent PARI-GP programs numerically tested the complete all-depth outputs through MZV weight 22 and Euler-sum weight 12.

9 Results

The data mine tests conjectured basis sizes and verifies explicit relations for Euler sums and MZVs across substantial weight and depth ranges. It also develops basis-selection procedures and generalized relations that support these checks.

  • The data mine uses its stored results to test conjectures about basis counts as functions of weight and depth.
  • Conjecture checks: The Zagier conjecture holds through weight 22, extends to weight 24 under a modular-calculus assumption, and to weight 26 under an additional depth assumption.
  • Conjecture checks: All Euler-sum runs agree with the Broadhurst conjecture, including complete verification through weight 12 and modular checks at higher weights and selected depths.For depth 4, complete verification reaches weight 22 and modular verification reaches weight 30.
  • Conjecture checks: MZV computations confirm the Broadhurst–Kreimer basis-size conjecture over a large range, including its minimal-depth Euler-sum representation.The second part requires corresponding Euler-sum results and is therefore harder to check.
  • Verified relations: Special Euler-sum and MZV relations, including relations mediated by alternating values, are verified analytically or through the data mine.Equation (9.19) implies a relation between MZVs mediated by one term with negative indices.
  • Basis selection: The proposed basis-selection procedure is non-unique because dependencies constrain which elements can be selected, so experimentation is required to obtain a suitable basis.The resulting bases are called pushdown bases, and constructions with the stated properties reach weight 26.
  • Pushdown enumeration: A new conjecture tracks basis elements by weight, depth, and pushdown, while predicting the first n-fold extension at weight w = 12n + 3 and depth d = 4n + 1.The formula has an exception for the first extension at weight 12.

10 Pushdowns

The paper interprets pushdowns as relations that express certain MZVs at smaller depth through Euler sums, and investigates their connection with doubling relations and A-functions. It constructs many such relations and proposes a basis framework incorporating them.

  • Pushdown is the phenomenon in which an MZV requiring a given depth can be represented through Euler sums at smaller depth.
  • Doubling relations: At weight 12, omitting generalized doubling relations leaves extra basis elements and no pushdown, whereas including them removes the extras and produces the pushdown.Using ordinary doubling without generalized doubling leaves one extra depth-4 element, but the pushdown still occurs.
  • A-functions: The A-function provides the non-MZV object in the pushdown representation and can sometimes be rewritten as MZVs, including one at depth d′ = d + 2.It contains half of the terms on the right-hand side of the doubling relation; in Z-notation these have an even number of negative indices.
  • Constructed relations: Pushdown relations were constructed for all extended basis elements through weight 21 and for one element at weight 22.Numerical LLL or PSLQ searches helped constrain the more difficult cases outside the data-mine files.
  • Scope: Not every A-function can be expressed solely in MZVs, and not every rewritable A-function yields a pushdown.Some rewritings use MZVs of no greater depth than the A-function itself.
  • Conjectured basis: The proposed conjecture selects either Z-values or A-values from Lyndon words, with the number of A-values fixed by the Broadhurst–Kreimer conjectures.Linear combinations of the selected A-values provide pushdowns for extensions by a pair of unit indices.

11 Special Euler Sums

The section examines when Euler sums and A-functions can be expressed using MZVs, including cases requiring higher-depth MZVs or pushdown objects.

  • MZV representations: The data mine tests which Euler sums and A-functions admit representations using MZVs only.The search distinguishes representations using MZVs of no greater depth from those requiring higher-depth MZVs.
  • MZV representations: A-functions may require MZVs of higher depth when an A-function already participates in a pushdown.Other A-functions can then be rewritten using that A-function and MZVs of the same or lower depth.
  • MZV representations: A large fraction of A-functions at affected depths can be rewritten entirely in terms of MZVs.At weight 17 and depth 5, 449 of 1365 finite A-functions have such representations.
  • Single-sum pushdowns: At weight 21 and depth 7, expressing pushdowns through a single Euler sum was difficult because each trial took several days.The search initially involved many candidates and required examining possible lower-weight Euler-sum products.
  • Single-sum pushdowns: Several candidate identities relate alternating-index Z-sums to H-functions, and an LLL search found a single-sum representation for Z3,−6,3,−6,3.The resulting pushdowns were obtained both in terms of MZVs and in terms of one Euler sum, although their index fields appear unrelated to the basis elements.

12 Outlook

The outlook identifies unresolved questions about generalized doubling relations, pushdowns, and basis construction, while emphasizing substantial computational barriers at higher weights and depths.

  • Open questions: Open questions concern simplifying generalized doubling relations and explaining their role in resolving leakage.The authors also ask why doubling relations are needed and how they relate to pushdowns.
  • Open questions: Further questions ask whether pushdown-relevant A-functions or pushdown bases can be identified without the full Euler-sum or MZV data mine.These questions target smaller subsets that could reduce computational requirements.
  • Computational limits: Partial evidence remains for double pushdowns at weights 27 and 28.At weight 27, a complete LLL formula would involve more than 800 elements and probably more than 10 times the current number of digits.
  • Computational limits: Higher-weight searches may require vastly greater resources, including Euler sums at weight 27 and depth 9.The authors suggest that algorithms using small subsets containing the A-functions could make these computations more tractable.
  • Continuation: The data mine will be extended as new relevant results become available, with additions and corrections recorded on a history page.The authors invite contributions from others.

A Fibonacci and Lyndon Bases at Fixed Weight

This section compares fixed-weight Fibonacci and Lyndon bases for MZVs and Euler sums, emphasizing alternative constructions and the depth-oriented bases used in the programs.

  • MZV bases: MZV bases can include Lyndon-word ζ-values and products of lower-weight ζ-values, with equivalent representations arising from different basis choices.The section discusses both Lyndon constructions and Hoffman-type alternatives.
  • MZV bases: The MZV Lyndon-basis size is conjectured through a Witt-type relation involving Perrin numbers.The recurrence is P1 = 0, P2 = 2, P3 = 3, and Pd = Pd−2 + Pd−3 for d ≥ 3.
  • Euler-sum bases: Euler-sum bases of Fibonacci type are counted by Fibonacci numbers and can be built from products of lower-weight basis elements.The section gives explicit low-weight examples involving ln(2), ζ-values, polylogarithms, and alternating sums.
  • Euler-sum bases: Alternative Euler-sum constructions use two-letter index alphabets and yield Padovan- or Fibonacci-counted bases.Some alternatives are not used because depth is not relevant to their counting or because they lack the desired depth orientation.
  • Euler-sum bases: The programs use a depth-oriented Fibonacci basis in which no element is reducible to lower-depth elements or products of lower-weight elements.The complete finite basis is provided through weight 12.

B Pushdown Bases

The pushdown bases are selected to organize primitive and extended elements while using A-functions suitable for pushdowns where the available data support that requirement.

  • Basis construction: The minimal pushdown basis selects maximal elements of Lw and minimal extended elements, with extended elements required to be Lyndon words.The construction also requires their corresponding A-functions to be usable for pushdowns up to weight 22.
  • Computational boundary: Two depth-9 elements are missing from one basis because of limited computer resources, although the construction based on L27 predicts them.The authors therefore describe the preceding basis as complete but the next one as computationally incomplete.
  • Computational boundary: At weight 28, too many elements are missing to provide a reliable basis list, despite available results with the leading depth omitted.A two-fold pushdown from depth 8 to depth 4 is nevertheless expected.

C Explicit pushdowns

The paper catalogs explicit pushdowns connecting MZV combinations to Euler-sum representations, with selected relations listed and complete formulas deposited in the data mine. These results expose patterns linked to doubling relations but remain computationally and structurally limited at higher depths and weights.

  • Explicit pushdowns: The listed pushdowns map combinations of MZVs at fixed weight and depth to Euler sums represented through H-, Z-, or A-function forms.The table gives explicit correspondences such as A7,5 → H−9,3 and Z−9,−3, alongside higher-depth examples.
  • Explicit pushdowns: The complete pushdown formulas are provided in the data mine because individual relations can contain up to about 150 terms.The programs section contains the complete formulas in the file pushdowns.h.
  • Explicit pushdowns: 2187 A7,5,3,5 +··· appears in one listed relation, while three others contain coefficients 337010625, 67402125, and 48144375 multiplying A11,5,5.The omitted terms are lower-depth MZVs or products of lower-weight MZVs.
  • Explicit pushdowns: 81 A7,5,3,3,3 +··· appears in another listed relation, with the ellipsis again denoting lower-depth MZVs or products of lower-weight MZVs.The first 15 relations used PSLQ and/or LLL, while seven could be derived with the data mine.
  • Limitations: The computational scope is limited: exact results reach weight 17 at depth 5 and weight 22 at depth 4, while the presented calculations used available resources to their limit.The authors also state that it is unclear whether the A-function scheme extends beyond the displayed pushdowns.
  • Explicit pushdowns: A-functions are preferred over single Euler sums because they contain half of the terms of the doubling relation and may reflect the origin of pushdowns.The authors report many possible single-Euler-sum choices but a unique A-function selection rule based on matching indices.
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