Linear independence of 1, $\mathrm{Li}_1$ and $\mathrm{Li}_2$

Abstract

We improve and extend the irrationality results proved by the authors (Ann. Sc. Norm. Super. Pisa Cl. Sci. $\left(5\right)$ 4:3 (2005), 389–437) for dilogarithms of positive rational numbers to results of linear independence over $\mathbb{Q}$ of $1$, ${Li}_1\left(x\right)$ and ${Li}_2\left(x\right)$ for suitable $x\in \mathbb{Q}$, both for $x>0$ and for $x<0$.