Generators for Vector Spaces Spanned by Double Zeta Values with Even Weight

TL;DR

This paper investigates the structure of vector spaces generated by double zeta values of even weight, providing explicit generators and bounds for their dimensions related to modular forms.

Contribution

It introduces specific generator sets for the quotient space of double zeta values, refining the understanding of their dimension bounds for even weights.

Findings

Explicit generator sets for the quotient space of double zeta values.

Upper bounds for the dimension of the space of double zeta values.

Connection between the dimension bounds and modular forms of weight k.

Abstract

Let $\mathcal{DZ}_k$ be the $\mathbb{Q}$-vector space spanned by double zeta values with weight $k$, and $\mathcal{DM}_k$ be its quotient space divided by the space $\mathcal{PZ}_k$ spanned by the zeta value $\zeta(k)$ and products of two zeta values with total weight $k$. When $k$ is even, an upper bound for the dimension of $\mathcal{DM}_k$ is known. By adding the dimensions of $\mathcal{DM}_k$ and $\mathcal{PZ}_k$, an upper bound of $\mathcal{DZ}_k$ which equals $k/2$ minus the dimension of the space of modular forms of weight $k$ on the modular group is given. In this note, we obtain some specific sets of generators for $\mathcal{DM}_k$ which represent the upper bound. These yield the corresponding sets and the upper bound for $\mathcal{DZ}_k$.