The Multiple Zeta Value Data Mine

Understand

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.