The $\mathbb{Z}$-module of multiple zeta values is generated by ones for indices without ones
Minoru Hirose, Takumi Maesaka, Shin-ichiro Seki, Taiki Watanabe
TL;DR
The paper proves that every multiple zeta value can be expressed as a $\mathbb{Z}$-linear combination of values with no entry equal to one, and provides an explicit algorithm for the expansion. Central to the method is the introduction of $\zeta^{\diamondsuit}$-values, which partially satisfy the same relations as ordinary MZVs, and an operator $\mathfrak{D}$ on the Hoffman algebra that yields integer expansion coefficients with respect to the basis $\mathbb{I}^{\geq 2}_k$; taking $N\to\infty$ passes from $\zeta^{\diamondsuit}$ to $\zeta$, giving explicit expressions $\zeta(\boldsymbol{k})=\sum_{\boldsymbol{l}\in\mathbb{I}^{\geq 2}_k} c_{\boldsymbol{k};\boldsymbol{l}}\zeta(\boldsymbol{l})$. The second main theme analyzes a class of relations, $\mathsf{Drop1}$, showing $\mathsf{LinKaw}^{*}\subset\mathsf{Drop1}$ via a finite-sum connected-sum framework (the F=G result), and clarifying how these relations interact with known Kawashima-type identities. The approach is elementary and finite-sum based, yielding a constructive pathway to Hoffman duality, duality phenomena, and explicit coefficient computation, and it connects classical MZV structure to dia-values and a concrete expansion algorithm with integer coefficients.
Abstract
We prove that every multiple zeta value is a $\mathbb{Z}$-linear combination of $ζ(k_1,\dots, k_r)$ where $k_i\geq 2$. Our proof also yields an explicit algorithm for such an expansion. The key ingredient is to introduce modified multiple harmonic sums that partially satisfy the relations among multiple zeta values and to determine the structure of the space generated by them.