#### Vol. 8, No. 1, 2019

 Recent Issues Volume 8, Issue 2 Volume 8, Issue 1
 The Journal About the Journal Subscriptions Editorial Board Submission Guidelines Submission Form Ethics Statement To Appear Editorial Login Contacts ISSN (electronic): 2640-7361 ISSN (print): 2220-5438 founded and published with the scientific support and advice of the Moscow Institute of Physics and Technology Other MSP Journals
Linear independence of 1, $\mathrm{Li}_1$ and $\mathrm{Li}_2$

### Georges Rhin and Carlo Viola

Vol. 8 (2019), No. 1, 81–96
DOI: 10.2140/moscow.2019.8.81
##### 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 $ℚ$ of $1$, ${Li}_{1}\left(x\right)$ and ${Li}_{2}\left(x\right)$ for suitable $x\in ℚ$, both for $x>0$ and for $x<0$.

##### Keywords
polylogarithms, linear independence measures, permutation group method, saddle-point method in $\mathbb{C}^2$
##### Mathematical Subject Classification 2010
Primary: 11J72
Secondary: 11J82, 33B30