## Abstract

Let $r,m$ be positive integers. Let $0\le x<1$ be a rational number. We denote by ${\mathrm{\Phi}}_{s}\left(x,z\right)$ the $s$-th Lerch function

$$\sum _{k=0}^{\infty}\frac{{z}^{k+1}}{{\left(k+x+1\right)}^{s}},$$

with $s=1,2,\dots ,r$. When $x=0$, this is the polylogarithmic function. Let ${\alpha}_{1},\dots ,{\alpha}_{m}$ be pairwise distinct algebraic numbers with $0<\left|{\alpha}_{j}\right|<1$ ($1\le j\le m$). We state a linear independence criterion over algebraic number fields of all the $rm+1$ numbers: ${\mathrm{\Phi}}_{1}\left(x,{\alpha}_{1}\right)$, ${\mathrm{\Phi}}_{2}\left(x,{\alpha}_{1}\right)$, $\dots $, ${\mathrm{\Phi}}_{r}\left(x,{\alpha}_{1}\right)$, ${\mathrm{\Phi}}_{1}\left(x,{\alpha}_{2}\right)$, ${\mathrm{\Phi}}_{2}\left(x,{\alpha}_{2}\right)$, $\dots $, ${\mathrm{\Phi}}_{r}\left(x,{\alpha}_{2}\right)$, $\dots $, ${\mathrm{\Phi}}_{1}\left(x,{\alpha}_{m}\right)$, ${\mathrm{\Phi}}_{2}\left(x,{\alpha}_{m}\right)$, $\dots $, ${\mathrm{\Phi}}_{r}\left(x,{\alpha}_{m}\right)$ and $1$. We obtain an explicit sufficient condition for the linear independence of values of the $r$ Lerch functions ${\mathrm{\Phi}}_{1}\left(x,z\right)$, $\dots $, ${\mathrm{\Phi}}_{r}\left(x,z\right)$ at $m$ *distinct* *points in an algebraic number field of arbitrary finite degree *without any assumptions on $r$ and $m$. When $x=0$, our result implies the linear independence of polylogarithms of distinct algebraic numbers of arbitrary degree, subject to a metric condition. We give an outline of our proof together with concrete examples of linearly independent polylogarithms.

## Citation

Sinnou David. Noriko Hirata-Kohno. Makoto Kawashima. "Can polylogarithms at algebraic points be linearly independent?." Mosc. J. Comb. Number Theory 9 (4) 389 - 406, 2020. https://doi.org/10.2140/moscow.2020.9.389