Vol. 89, No. 2, 1980

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Integral formulas and integral tests for series of positive matrices

Audrey Anne Terras

Vol. 89 (1980), No. 2, 471–490

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.

Mathematical Subject Classification
Primary: 10D24, 10D24
Secondary: 10E35, 10D20, 10C15
Received: 2 February 1979
Published: 1 August 1980
Audrey Anne Terras