Can polylogarithms at algebraic points be linearly independent?

Abstract

Let $r,m$ be positive integers. Let $0\le x <1$ be a rational number. Let $Φ_s(x,z)$ be the $s$-th Lerch function $\sum_{k=0}^{\infty}\tfrac{z^{k+1}}{(k+x+1)^s}$ with $s=1,2,\ldots ,r$. When $x=0$, this is the polylogarithmic function. Let $α_1,\ldots ,α_m$ be pairwise distinct algebraic numbers with $0<|α_j|<1$ $(1 \le j \le m)$. In this article, we state a linear independence criterion over algebraic number fields of all the $rm+1$ numbers $:$ $Φ_1(x,α_1),Φ_2(x,α_1),\ldots, Φ_r(x,α_1),Φ_1(x,α_2),Φ_2(x,α_2),\ldots, Φ_r(x,α_2),\ldots,Φ_1(x,α_m),Φ_2(x,α_m),\ldots, Φ_r(x,α_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the $r$ Lerch functions $Φ_1(x,z),Φ_2(x,z),\ldots, Φ_r(x,z)$ at $m$ distinct algebraic points without any assumption for $r$ and $m$, even for the case $x=0$, the polylogarithms. We give an outline of our proof and explain basic idea.