Odd zeta motive and linear forms in odd zeta values

TL;DR

This paper explores a family of mixed Tate motives over integers, providing integral formulas and geometric insights into linear forms involving odd zeta values, connecting motives, integrals, and series representations.

Contribution

It introduces a new geometric construction of mixed Tate motives that encapsulate all non-trivial extensions, linking integrals, motives, and series representations of zeta values.

Findings

Derived a general integral formula for coefficients of linear forms in zeta values.

Provided a geometric interpretation for the vanishing of certain coefficients.

Established compatibility between motives and series representations of periods.

Abstract

We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $\zeta(n)$. They naturally include the Beukers-Rhin-Viola integrals for $\zeta(2)$ and the Ball-Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over $\mathbb{Z}$ which contains all the non-trivial extensions between simple objects. In a joint appendix with Don Zagier, we prove the compatibility between the structure of the motives considered here and the representations of their periods as sums of series.