On relations between homogeneous spaces and the Riemann zeta-function

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The relation between the Riemann zeta-function and the distribution of prime numbers was traditionally discussed in view of their possible direct interaction with the Riemann Hypothesis ; thus the stress of the research was laid upon the qualitative aspect of the zeta-function. The aim of our research is, however, to shift our attention to the quantitative aspect of this fundamental function. The feasibility of such an argument is indicated, for instance, by the well-known fact that as far as the distribution of prime numbers in short intervals is concerned the Riemann Hypothesis might be replaced by a certain quantitative property of the zeta-function. The latter is the moment problem of the values of the zeta-function along the critical line. It had been discussed with purely classical means until we recently succeeded in establishing its relation with the spectral resolution of the hyperbolic Laplacian, exhibiting in particular the possibility of a new view point that the Riemann ze … More ta-function might be taken for a generator of Hecke L-functions (Maass waves). In other words it can be said that the quantitative nature of the zeta-function contains some wave components that stand for a structure of the hyperbolic plane. In our research we tried to extend and refine our findings. To this end we employed two methods. One was to appeal to the theory of the trace-formulas for groups of higher rank in order to analyze the problem of higher power moments of the zeta-function. The other was to exploit the finer group structure of the full modular group in order to get a more refined image of the zeta-function. Along the former line we could extract a fact that strongly suggested a possibility of an essential role playd by the SL (3, Z) trace-formula in the theory of the 6th power moment problem. It should, however, be stressed that we found also that contrary to what had been expected the 8th power moment problem could be reduced to the theory of SL (2, Z). This finding seems to indicate a new prospect of the theory of power moments for the zeta-function. As for the research along the second line we report that we could establish a close relation between the Riemann zeta-function and Hecke congruence subgroups. It should be worth remarking that we found that Selbergs eigen-value problem could be discussed in the frame of the theory of the power moments of the zeta-function. Less