The Multiple Zeta Value Data Mine (0907.2557v2)

Abstract: 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.

• The paper introduces extensive proven reductions of multiple zeta values (w ≤ 22) and Euler sums (w ≤ 12) to bases aligning with established conjectures.
• It applies modular arithmetic and the Generalized Doubling Relations to extend analysis of these sums up to weight 30.
• Leveraging FORM-based algebra, the study verifies large-scale computations and lays the groundwork for advanced quantum field theory research.

Overview of Multiple Zeta Value Data Mine

The paper "The Multiple Zeta Value Data Mine" by J. Blumlein, D.J. Broadhurst, and J.A.M. Vermaseren presents a comprehensive exploration of multiple zeta values (MZVs) and Euler sums. These sums have a significant role in both mathematics and physics, particularly in perturbative quantum field theory. The paper elaborates on a data mine of proven results for MZVs with weights up to 22 and Euler sums with weights up to 12, achieved by using computational algebra techniques.

Key Contributions

1. Explicit Proven Reductions: The authors have successfully reduced all MZVs with weights $w \leq 22$ and all Euler sums with weights $w \leq 12$ to bases that align with the Broadhurst–Kreimer and Broadhurst conjectures. These bases are graded by weight and depth.
2. Modular Arithmetic for Higher Weights: The paper extends the exploration to greater weights (up to 30) using modular arithmetic, providing additional support for the conjectures. The authors have introduced the Generalized Doubling Relations that facilitate deriving new relations for Euler sums.
3. Pushdown Mechanism: The authors present the "pushdown" mechanism, which simplifies the enumeration of primitive Euler sums. This mechanism implies that some aspects of MZVs can be related to simpler structures, possibly originating from doubling relations.
4. Computational Approach: The research leverages the FORM computer algebra language to generate these results. The data mine produced could be instrumental in further analyzing the double-shuffle algebra of MZVs and their Euler counterparts.

Numerical Results and Verification

The authors introduce a new computational technique involving FORM and TFORM, allowing them to handle extremely large and complex algebraic computations. The correctness of the generated data is verified by matching the derived basis sizes with those predicted by conjectures, confirming the computational accuracy of their methodologies.

Implications and Future Directions

The paper enriches the comprehension of MZVs and Euler sums by providing a structured dataset and methodology that can be utilized for further theoretical and practical inquiries. The findings have immediate implications for quantum field theory, notably in higher-loop calculations. Additionally, this work lays a foundation for evaluating more sophisticated algebraic structures involved in calculating complex Feynman diagrams.

Going forward, the authors speculate on potential expansions of the data mine, particularly with advances in computational resources and algorithms. This research opens the door to questions such as exploring deeper connections within the algebraic structures of MZVs and examining their implications in other areas of mathematics and physics.

Conclusion

In summary, "The Multiple Zeta Value Data Mine" provides an expansive dataset and computational framework for studying multiple zeta values and Euler sums, substantiated by rigorous computational techniques. This paper not only verifies various longstanding conjectures but also facilitates new explorations into the algebraic structures governing these mathematical entities. The paper is a significant contribution to the field, enhancing both the theoretical understanding and the computational tools available for future research.