On the Riemann zeta-function and the 3D hyperbolic variety

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Our research was a continuation of our former project (grant-in-aid no.07640072). The latter had been devoted to the investigation of a relation between the Riemann zeta-function and automorphic forms of one complex variable, i.e., functions on the upper half-plane which are automorphic with respect to certain arithmetically defined fuchsian groups, especially the modular group and its Hecke congruence subgroups. The aim of the project under summarizing was to extend such a relation to a wider class of zeta-functions and automorphlic functions living in higher dimensional varieties which are, respectively, natural generalizations of the Riemann zeta-function and the hyperbolic upper half-plane. As a first step of our grand project, we took up, in this project, the case of the Dedekind zeta-function of the Gaussian number field. This choice was made not only for the sake of structural simplicity but because of the fact that it raises a genuine new situation much different from that we experienced in our former project. More precisely, we found, with the co-operation of Prof. Bruggeman of Utrecht University, that in the quest of such an extension we need to move to the Lie group SL (2, C) and its unitary representations. In that way we have established an extension of Kuznetsov's sum formula to the complex situation, and, as a consequence, a spectral decomposition of the fourth power moment of the Dedekind zeta-function of the Gaussian number field. These accomplishments fulfill the aim of our project.