Motivic double zeta values of odd weight

TL;DR

This paper establishes a short exact sequence for motivic double zeta values of odd weight, classifies their relations, and confirms Zagier's conjecture on the rank of a related matrix, advancing understanding of multiple zeta values.

Contribution

It introduces a new exact sequence for motivic double zeta values of odd weight and classifies all their relations, confirming a conjecture of Zagier.

Findings

Established a short exact sequence for motivic double zeta values.

Classified all relations among depth-graded motivic double zeta values.

Confirmed Zagier's conjecture on the rank of a matrix related to odd weight multiple zeta values.

Abstract

For odd $N\geq 5$, we establish a short exact sequence about motivic double zeta values $\zeta^{\mathfrak{m}}(r,N-r)$ with $r\geq3$ odd, $N-r\geq2$. From this we classify all the relations among depth-graded motivic double zeta values $\zeta^{\mathfrak{m}}(r,N-r)$ with $r\geq3$ odd, $N-r\geq2$. As a corollary, we confirm a conjecture of Zagier on the rank of a matrix which concerns relations among multiple zeta values of odd weight.