Can polylogarithms at algebraic points be linearly independent?

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.