  In this paper, new probability bounds are derived for algebraic lattice
codes. This is done by using the Dedekind zeta functions of the algebraic
number fields involved in the lattice constructions. In particular, it is shown
how to upper bound the error performance of a finite constellation on a
Rayleigh fading channel and the probability of an eavesdropper's correct
decision in a wiretap channel. As a byproduct, an estimate of the number of
elements with a certain algebraic norm within a finite hyper-cube is derived.
While this type of estimates have been, to some extent, considered in algebraic
number theory before, they are now brought into novel practice in the context
of fading channel communications. Hence, the interest here is in
small-dimensional lattices and finite constellations rather than in the
asymptotic behavior.
