Multiple zeta values in positive characteristic break their predictable patterns at weight 2q+1, shattering a 15-year-old rule.
April 29, 2026
Original Paper
The threshold for linear independence of multiple zeta values in positive characteristic
arXiv · 2604.25451
The Takeaway
Multiple zeta values are mathematical constants that describe deep links between number theory and particle physics. These values were previously assumed to stay linearly independent as they grew larger. This proof identifies a hidden relationship that only emerges once the weight reaches a threshold of 2q+1. The discovery provides the first counterexample to a conjecture that guided the field for 15 years. It reveals that the rules governing these numbers are far more restrictive than anyone thought. This shift forces a total rethink of how arithmetic functions behave in complex systems.
From the 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 $\