  Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x
<1$. Consider $\Phi_s(x,z)
=\displaystyle\sum_{k=0}^{\infty}\frac{z^{k+1}}{{(k+x+1)}^s}$ the $s$-th Lerch
function with $s=1, 2, \cdots, r$. When $x=0$, this is a polylogarithmic
function. Let $\alpha_1, \cdots, \alpha_m$ be pairwise distinct algebraic
numbers of arbitrary degree over the rational number field, with
$0<|\alpha_j|<1 \,\,\,(1\leq j \leq m)$. In this article, we show a criterion
for the linear independence, over an algebraic number field containing
$\mathbb{Q}(\alpha_1, \cdots, \alpha_m)$, of all the $rm+1$ numbers :
$\Phi_1(x,\alpha_1)$, $\Phi_2(x,\alpha_1), $ $\cdots , \Phi_r(x,\alpha_1)$,
$\Phi_1(x,\alpha_2)$, $\Phi_2(x,\alpha_2), $ $\cdots , \Phi_r(x,\alpha_2),
\cdots, \cdots, \Phi_1(x,\alpha_m)$, $\Phi_2(x,\alpha_m)$, $\cdots ,
\Phi_r(x,\alpha_m)$ and $1$. This is the first result that gives a sufficient
condition for the linear independence of values of the Lerch functions at
several distinct algebraic points, not necessarily lying in the rational number
field nor in quadratic imaginary fields. We give a complete proof with
refinements and quantitative statements of the main theorem announced in [10],
together with a proof in detail on the non-vanishing Wronskian of Hermite type.
