Let
$r,m$ be positive integers.
Let
$0\le x<1$ be a rational
number. We denote by
${\Phi}_{s}\left(x,z\right)$
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:
${\Phi}_{1}\left(x,{\alpha}_{1}\right)$,
${\Phi}_{2}\left(x,{\alpha}_{1}\right)$,
$\dots \phantom{\rule{0.3em}{0ex}}$,
${\Phi}_{r}\left(x,{\alpha}_{1}\right)$,
${\Phi}_{1}\left(x,{\alpha}_{2}\right)$,
${\Phi}_{2}\left(x,{\alpha}_{2}\right)$,
$\dots \phantom{\rule{0.3em}{0ex}}$,
${\Phi}_{r}\left(x,{\alpha}_{2}\right)$,
$\dots \phantom{\rule{0.3em}{0ex}}$,
${\Phi}_{1}\left(x,{\alpha}_{m}\right)$,
${\Phi}_{2}\left(x,{\alpha}_{m}\right)$,
$\dots \phantom{\rule{0.3em}{0ex}}$,
${\Phi}_{r}\left(x,{\alpha}_{m}\right)$ and
$1$. We
obtain an explicit sufficient condition for the linear independence of values of the
$r$ Lerch
functions
${\Phi}_{1}\left(x,z\right)$,
$\dots \phantom{\rule{0.3em}{0ex}}$,
${\Phi}_{r}\left(x,z\right)$ 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.
