The threshold for linear independence of multiple zeta values in positive characteristic

TL;DR

This paper determines the exact weight threshold for linear independence of multiple zeta values in positive characteristic, confirming the conjecture up to a point and providing a counterexample at the threshold.

Contribution

It establishes the precise weight threshold for linear independence of MZVs in positive characteristic and introduces new methods connecting MZVs with Carlitz polylogarithms.

Findings

Linear independence holds for weights up to 2q.

At weight 2q+1, a unique explicit relation exists.

First counterexample to Thakur's conjecture is provided.

Abstract

A fundamental conjecture formulated by Thakur in 2009, which has guided significant developments in function field arithmetic, asserts that multiple zeta values (MZV's) in positive characteristic of fixed weight are linearly independent over $\mathbb{F}_q$. In this paper we settle this conjecture by determining the precise threshold for this independence. We prove that linear independence holds for all weights up to 2q, while for weight 2q+1 we establish the existence of a unique and explicit $\mathbb{F}_q$-linear relation. This result provides the first counterexample to Thakur's conjecture. Our proof relies on a new connection between MZVs and Carlitz multiple polylogarithms over $\mathbb{F}_q$, generalizing a central result of [IKLNDP24]. We also introduce a modification of the algorithm from [ND21] that yields a weight-preserving operator acting on $\mathbb{F}_q$-linear relations, providing the algebraic framework for these results.