Conjecture 3.16 — Zagier

Definitions and mathematical statement

For an admissible tuple of integers $(s_1,\ldots,s_r)$, meaning $s_1\ge2$ and $s_j\ge1$ for $j>1$, define \[ \zeta(s_1,\ldots,s_r)=\sum_{n_1>\cdots>n_r\ge1} n_1^{-s_1}\cdots n_r^{-s_r}. \] The series converges, and the tuple's weight is $s_1+\cdots+s_r$. Let $\mathfrak Z_w$ be the $\mathbb Q$-linear span in $\mathbb R$ of the values of weight $w$, with $\mathfrak Z_0=\mathbb Q$ and $\mathfrak Z_1=\{0\}$. Put $d_w=\dim_{\mathbb Q}\mathfrak Z_w$.

Zagier's dimension conjecture asks whether the exact dimensions satisfy \[ d_0=1,\quad d_1=0,\quad d_2=1,\qquad d_w=d_{w-2}+d_{w-3}\quad(w\ge3). \] Equivalently, their formal generating series should be $\sum_{w\ge0}d_w T^w=(1-T^2-T^3)^{-1}$.

There are only finitely many admissible tuples of a fixed weight, so each space is finite dimensional. Its dimension counts rationally independent actual real values, not the number of index tuples or of formal symbols before relations are imposed. Values from different tuples can coincide or satisfy rational linear identities.

The target is equality for every weight, requiring both sufficiently many independent values and enough relations to give the stated dimension. Proving only one inequality against the recursive sequence does not establish the conjecture. The question concerns each fixed-weight space and does not, by itself, assert independence between spaces of different weights.

Short English statement

Are the exact rational dimensions of the spaces of multiple zeta values given by Zagier’s weight recurrence?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 8700034 (AMR-086-0034), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Waldschmidt - Open Diophantine Problems (2004), Conjecture 3.16. Original source indexed by AMR-086-0034. https://arxiv.org/abs/math/0312440.
• S3 M. Waldschmidt, Open Diophantine Problems, Moscow Mathematical Journal 4 (2004), 245–305. Author PDF supplies the surrounding definitions and hypotheses. https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/odp.pdf.