  We prove that the number of Siegel-reduced bases for a randomly chosen
$n$-dimensional lattice becomes, for $n \rightarrow \infty$, tightly
concentrated around its mean. We also show that most reduced bases behave as in
the worst-case analysis of lattice reduction.
  Comparing with experiment, these results suggest that most reduced bases
will, in fact, "very rarely" occur as an output of lattice reduction. The
concentration result is based on an analysis of the spectral theory of
Eisenstein series and uses (probably in a removable way) the Riemann
hypothesis.

Number Theory