The linear independence of $1$, $\zeta(2)$, and $L(2,\chi_{-3})$

TL;DR

This paper proves the irrationality and linear independence over Q of 1, ζ(2), and L(2,χ_{-3}), using a novel arithmetic holonomy bound applied to Zagier's construction, with broader applications in irrationality proofs.

Contribution

It introduces a new arithmetic holonomy bound technique to establish the linear independence of specific special values, advancing irrationality proof methods.

Findings

Proves the irrationality of L(2,χ_{-3})

Establishes Q-linear independence of 1, ζ(2), and L(2,χ_{-3})

Applies the method to other irrationality problems

Abstract

We prove the irrationality of the classical Dirichlet L-value $L(2,\chi_{-3})$. The argument applies a new kind of arithmetic holonomy bound to a well-known construction of Zagier. In fact our work also establishes the $\mathbf{Q}$-linear independence of $1$, $\zeta(2)$, and $L(2,\chi_{-3})$. We also give a number of other applications of our method to other problems in irrationality.