Motivic multiple zeta values reletive to \mu_2

TL;DR

This paper studies motivic multiple zeta values relative to _2, establishing bases and exact sequences, and proves conjectures related to sum odd double zeta values and their analogues in higher depth.

Contribution

It provides new bases and exact sequences for motivic double and triple zeta values relative to _2, and proves several conjectures by Kaneko and Tasaka.

Findings

Established a short exact sequence for depth-graded motivic double zeta values.

Found bases for depth-graded motivic double and triple zeta values relative to _2.

Proved conjectures about sum odd double zeta values and their analogues in depth three.

Abstract

We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to $\mu_2$. We find a basis for the depth-graded motivic double zeta values relative to $\mu_2$ of even weight and a basis for the depth-graded motivic triple zeta values relative to $\mu_2$ of odd weight. As an application of our main results, we prove Kaneko and Tasaka's conjectures about the sum odd double zeta values and the classical double zeta values. We also prove an analogue of Kaneko and Tasaka's conjecture in depth three. At last we formulate a conjecture which is related to sum odd multiple zeta values in higher depth.