On zeta values in Tate algebras

The so-called Carlitz zeta values,
first considered by Carlitz in the
years 1930, belong to local fields
in positive characteristic and
share several properties with the
more classical values of
Riemann's zeta function
at integers greater than one.
After a short introduction, we
will introduce zeta values in Tate
algebras. These will be strange
hybrids between "special values"
and "special functions". We will
sketch a picture of what we know
about them.