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We present the first example of the Selberg type zeta function for noncompacthigher rank locally symmetric spaces. We study certain Selberg type zetafunctions and Ruelle type zeta functions attached to the Hilbert modular groupof a real quadratic field. We show that they have meromorphic extensions to thewhole complex plane and satisfy functional equations. The method is based onconsidering the differences among several Selberg trace formulas with differentweights for the Hilbert modular group. Besides as an application of thedifferences of the Selberg trace formula, we also obtain an asymptotic averageof the class numbers of indefinite binary quadratic forms over the realquadratic integer ring.

Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces