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Hoffman basis conjecture for multiple zeta values

Let Z\mathcal{Z} be the Q\mathbb{Q}-vector space of multiple zeta values, and let

H=span⁡Q{ζ(k1,…,kd)∣ki∈{2,3} for i=1,…,d}⊂Z.\mathcal{H}=\operatorname{span}_{\mathbb{Q}}\{\zeta(k_1,\ldots,k_d)\mid k_i\in\{2,3\}\text{ for }i=1,\ldots,d\}\subset\mathcal{Z}.

Hoffman basis conjecture. The Hoffman elements ζ(k1,…,kd)\zeta(k_1,\ldots,k_d) with ki∈{2,3}k_i\in\{2,3\} form a basis for Z\mathcal{Z}; in particular,

H=Z.\mathcal{H}=\mathcal{Z}.

The conjecture proposes an explicit basis indexed by words in 22 and 33. The source attributes it to Hoffman; its general status is open.

Sources & referencesView supporting material

Primary source

Annika Burmester, Niclas Confurius and Ulf Kühn, “AGZT-Lectures on formal multiple zeta values”, arXiv:2406.13630 (2024).

Additional references

8 papers in this index state this conjecture (2006–2024). The statement above is taken from the most recent of them; the others are arXiv:2402.11539, arXiv:2301.05906, arXiv:2205.07165, arXiv:1407.5165, arXiv:0712.1656, arXiv:math/0609799, arXiv:math/0601151.

multiple-zeta-valuesperiods

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