Let
be positive integers.
Let
be a rational
number. We denote by
the
-th
Lerch function
with
. When
, this is the polylogarithmic
function. Let
be pairwise
distinct algebraic numbers with
(). We
state a linear independence criterion over algebraic number fields of all the
numbers:
,
,
,
,
,
,
,
,
,
,
,
,
and
. We
obtain an explicit sufficient condition for the linear independence of values of the
Lerch
functions
,
,
at
distinctpoints in an algebraic number field of arbitrary finite degree without any assumptions
on
and
.
When
,
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.
Dedicated to the memory of Professor
Naum Ilyitch Feldman
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