The main results of this
paper are integral formulas which generalize that used by Siegel to prove the
Minkowski-Hlawka theorem in the geometry of numbers. The main application is the
derivation of an integral test for Dirichlet series of several complex variables defined
by sums over integer matrices. Such an integral test yields an easy proof of the
convergence of Eisenstein series, whose analytic continuations are important
in harmonic analysis on Minkowski’s fundamental domain for the positive
n × n real matrices modulo n × n integer matrices of determinant ±1 (i.e.,
O(n) ∖ GL(n,R)∕GL(n,Z)). These integral tests can also be used to analyze the
analytic continuation of Eisenstein series as sums of higher dimensional incomplete
gamma functions.
