What question did this study set out to answer?

This work aims to show that every multiple zeta value can be generated as a linear combination of a specific type of zeta value.

December 21, 2025Open Access

The Z-module of multiple zeta values is generated by ones for indices without ones

Key Points

This work aims to show that every multiple zeta value can be generated as a linear combination of a specific type of zeta value.
Introduced modified multiple harmonic sums
Established relations among multiple zeta values
Determined the structure of the generated space
Every multiple zeta value is expressible as a linear combination of zeta(k_1,...,k_r) where k_i≥2
An explicit algorithm for such expansions was developed

Abstract

We prove that every multiple zeta value is a Z-linear combination of (k₁, , kᵣ) where kᵢ 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.

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