  In this paper we show how to use elementary methods to prove that the volume
of Sl_k R / Sl_k Z is zeta(2) * zeta(3) * ... * zeta(k) / k. Using a version of
reduction theory presented in this paper, we can compute the volumes of certain
unbounded regions in Euclidean space by counting lattice points and then appeal
to the machinery of Dirichlet series to get estimates of the growth rate of the
number of lattice points appearing in the region as the lattice spacing
decreases.
  We also present a proof of the closely related result that the Tamagawa
number of Sl_k Q is 1 that is somewhat simpler and more arithmetic than Weil's.
His proof proceeds by induction on k and appeals to the Poisson summation
formula, whereas the proof here brings to the forefront local versions of the
formula, one for each prime p, which help to illuminate the appearance of
values of zeta functions in formulas for volumes.

Number Theory;K-Theory and Homology