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
PDF Access Denied
We have not been able to recognize your IP address
3.144.25.130
as that of a subscriber to this journal.
Online access to the content of recent issues is by
subscription, or purchase of single articles.
Please contact your institution's librarian suggesting a subscription, for example by using our
journal-recommendation form.
Or, visit our
subscription page
for instructions on purchasing a subscription.