  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
$\sum_{k=0}^{\infty}\tfrac{z^{k+1}}{(k+x+1)^s}$ with $s=1,2,\ldots ,r$. When
$x=0$, this is the polylogarithmic function. Let $\alpha_1,\ldots ,\alpha_m$ be
pairwise distinct algebraic numbers with $0<|\alpha_j|<1$ $(1 \le j \le m)$. In
this article, we state a linear independence criterion over algebraic number
fields of all the $rm+1$ numbers $:$
$\Phi_1(x,\alpha_1),\Phi_2(x,\alpha_1),\ldots,
\Phi_r(x,\alpha_1),\Phi_1(x,\alpha_2),\Phi_2(x,\alpha_2),\ldots,
\Phi_r(x,\alpha_2),\ldots,\Phi_1(x,\alpha_m),\Phi_2(x,\alpha_m),\ldots,
\Phi_r(x,\alpha_m)$ and $1$. This is the first result that gives a sufficient
condition for the linear independence of values of the $r$ Lerch functions
$\Phi_1(x,z),\Phi_2(x,z),\ldots, \Phi_r(x,z)$ 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.
