Can polylogarithms at algebraic points be linearly independent?

Sinnou David, Noriko Hirata-Kohno, Makoto Kawashima

Let $r,m$ be positive integers. Let $0\le x<1$ be a rational number. Let ${\Phi }_s(x,z)$ be the $s$-th Lerch function ${\sum }_{k=0}^{\infty }\frac{z^{k+1}}{(k+x+1{)}^s}$ with $s=1,2,\ldots…