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

我们的研究是我们以前的项目（资助号07640072）的延续。后者致力于研究黎曼ζ函数与单复变量的自同构形式之间的关系，即上半平面上对某些算术定义的fuchsian群，特别是模群及其Hecke同余子群自同构的函数。总结项目的目的是将这种关系扩展到更广泛的ζ函数和高维自同态函数，它们分别是黎曼ζ函数和双曲上半平面的自然推广。作为我们这个大项目的第一步，我们在这个项目中，研究了高斯数场的Dedekind ζ函数。这种选择不仅是为了结构的简单性，而且因为它提出了一个真正的新情况，这与我们在以前的项目中所经历的大不相同。更准确地说，在乌得勒支大学的Bruggeman教授的合作下，我们发现，在寻求这种扩展时，我们需要转向李群SL （2， C）及其酉表示。通过这种方式，我们建立了库兹涅佐夫求和公式在复杂情况下的推广，并由此得到高斯数场的Dedekind ζ函数的第四次幂矩的谱分解。这些成就实现了我们项目的目标。

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.