The $\mathbb{Z}$-module of multiple zeta values is generated by ones for indices without ones

TL;DR

This paper proves that all multiple zeta values can be expressed as integer linear combinations of those with indices greater than or equal to two, providing an explicit algorithm for such expansions.

Contribution

It introduces modified multiple harmonic sums and determines the structure of the generated space, establishing a basis for multiple zeta values without ones.

Findings

Every multiple zeta value is a $bZ$-linear combination of those with indices $ eq 1$.

An explicit algorithm for expressing multiple zeta values without ones.

Structural understanding of the space generated by modified multiple harmonic sums.

Abstract

We prove that every multiple zeta value is a $\mathbb{Z}$-linear combination of $\zeta(k_1,\dots, k_r)$ 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.