Period polynomial relations between formal double zeta values of odd weight

TL;DR

This paper establishes new relations between double zeta values of odd weight, linking them to cusp forms and addressing questions about the structure of these values, extending previous even-weight results.

Contribution

It provides a formula for relations between double zeta values of odd weight and connects these to the space of cusp forms, extending known results to odd weights.

Findings

Derived relations between double zeta values of odd weight.

Connected double zeta relations to cusp form spaces.

Answered a question by Zagier regarding the kernel of a related matrix.

Abstract

For odd $k$, we give a formula for the relations between double zeta values $\zeta(r,k-r)$ with $r$ even. This formula provides a connection with the space of cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. This is the odd weight analogue of a result by Gangl, Kaneko and Zagier. We also provide an answer of a question asked by Zagier about the left kernel of some matrix. Although the restricted sum statement fails in the odd weight case, we provide an asymptotical statement that replaces it. Our statement works more generally for restricted sums with any congruence condition on the first entry of the double zeta value.