Relation between the Riemann zeta-function and the Casimir operator

我们研究的主要目的是发展一种理论，从李群的几何角度来把握黎曼zeta函数，而不需要Kloosterman和的理论。我们与R.W.布鲁格曼实现了这一目标。我们可以把λ（s）的四次幂矩作为某个Poincare级数的一个特殊值嵌入到空间L^2（PSL_2（Z）\PSL_2R）中，并以一种新的方式恢复其已知的谱分解，即，完全不依赖Kloosterman求和理论。我们的这项工作也揭示了一个深刻的关系之间的（s）和Jacquet-Langlands局部函数方程。这一事实将可能使我们的理论扩展到更高维的情况下，否则几乎是不可能的。我们的结果可以被评价为zeta函数理论的一个基本成就。

The principal purpose of our research was to develop a theory to grasp the Riemann zeta-function ζ(s) in the geometric perspective of Lie groups, dispensing with the theory of Kloosterman sums. We, under a joint research activity with R.W. Bruggeman, achieved this aim. We could embed the fourth power moment of ζ(s) in the space L^2(PSL_2(Z)\PSL_2R) as a special value of a certain Poincare series, and retrieved its known spectral decomposition in a new way, i.e., completely without recourse to the theory of Kloosterman sums. This work of ours revealed also a deep relation between ζ(s) and the Jacquet-Langlands local functional equation. This fact will possibly make it feasible to extend our theory to higher dimensional situation, which otherwise should have been almost impossible. Our result could be assessed as to be a basic achievement in the theory of zeta-functions.