Integral formulas and integral tests for series of positive matrices

Abstract

The main results of this paper are integral formulas which generalize that used by Siegel to prove the Minkowski-lawka 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 nxn real matrices modulo nxn integer matrices of determinant 1 (i.e., O(n)\GL{n, R)/GL(n t Z)). These integral tests can also be used to analyze the analytic continuation of Eisenstein series as sums of higher dimensional incomplete gamma functions.