Todd’s relation conjecture

Todd's relation conjecture asserts that all $\mathbb{F}_q(\theta)$ -linear relations among Thakur's multiple zeta values in positive characteristic are generated from the fundamental binary relation by the operators $\mathcal{B}^{\mathrm{S}}$ , $\mathcal{C}^{\mathrm{S}}$ , and $\mathcal{B}^{\mathrm{S}}\circ\mathcal{C}^{\mathrm{S}}$ . The source also states that the same relations are generated by $\mathcal{B}^{\ast\mathrm{S}}$ , $\mathcal{C}^{\mathrm{S}}$ , and $\mathcal{B}^{\ast\mathrm{S}}\circ\mathcal{C}^{\mathrm{S}}$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Todd's relation conjecture via the starred operator
The same $\mathbb{F}_q(\theta)$ -linear relations among Thakur's multiple zeta values are generated from the fundamental binary relation by $\mathcal{B}^{\ast\mathrm{S}}$ , $\mathcal{C}^{\mathrm{S}}$ , and $\mathcal{B}^{\ast\mathrm{S}}\circ\mathcal{C}^{\mathrm{S}}$ .
source: Hu, Jinyuan, "Todd's relation conjecture and binary relations for multiple zeta values in positive characteristic"

References

Primary source

arXiv

Additional references

Todd's relation conjecture and binary relations for multiple zeta values in positive characteristic — arXiv

multiple-zeta-values number-theory positive-characteristic-arithmetic

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

An unrefereed 2026 preprint claims to prove the conjecture, but the result has not been independently checked.

Todd’s relation conjecture predicts a complete structural description of all relations among Thakur’s multiple zeta values in positive characteristic.

Known results

A 2022 preprint claims to determine all linear relations among Thakur’s multiple zeta values and related values, including an explicit basis for each weight.
A second 2022 preprint claims to prove Thakur’s basis conjecture and derive Todd’s dimension conjecture, with a mechanism for all fixed-weight relations using Carlitz multiple polylogarithms.

September 2026 claimed proof

A September 8, 2026 report cites the preprint Todd's relation conjecture and binary relations for multiple zeta values in positive characteristic, which claims the full conjecture for Thakur’s multiple zeta values and related results for Carlitz multiple polylogarithms and generating functions. The claim is unrefereed and has no independent confirmation in the retrieved sources.

Current status (as of September 2026): A full proof is claimed in an unrefereed preprint, but Todd’s relation conjecture remains unverified.

Sources

No solutions have been posted yet.