Mixed motives and linear forms in the Catalan constant

TL;DR

This paper constructs a geometric mixed motive over rationals with the Catalan constant as a period and uses it to generate explicit linear forms involving 1 and the Catalan constant.

Contribution

It introduces a new geometric construction of a mixed motive related to the Catalan constant and derives explicit linear forms in 1 and G from this motive.

Findings

Constructed a 2-dimensional mixed motive over Q with G as a period

Derived explicit linear forms in 1 and G from the motive

Computed coefficients of 1 and G in these linear forms

Abstract

We first give a geometric construction of a 2-dimensional mixed motive over $\mathbb{Q}$ with the Catalan constant $\mathbf{G}=1-1/3^2+1/5^2-1/7^2+\cdots$ as a period. We then use this motive to obtain a supply of linear forms in 1 and $\mathbf{G}$. We also explicitly compute the coefficients of 1 and $\mathbf{G}$ in these linear forms.