On relations between homogeneous spaces and the Riemann zetafunction
Project/Area Number 
05804004

Research Category 
GrantinAid for General Scientific Research (C)

Allocation Type  Singleyear Grants 
Research Field 
Algebra

Research Institution  Nihon University 
Principal Investigator 
MOTOHASHI Yoichi Nihon University, College of Science of Technology, Professor, 理工学部, 教授 (30059969)

Project Period (FY) 
1993 – 1994

Project Status 
Completed (Fiscal Year 1994)

Budget Amount *help 
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 1994: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1993: ¥1,000,000 (Direct Cost: ¥1,000,000)

Keywords  ZetaFunction / Homogeneous Spaces / Distribution of Primes / 保型形式 / 整数論 
Research Abstract 
The relation between the Riemann zetafunction 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 zetafunction. 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 wellknown 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 zetafunction. The latter is the moment problem of the values of the zetafunction 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
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tafunction might be taken for a generator of Hecke Lfunctions (Maass waves). In other words it can be said that the quantitative nature of the zetafunction 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 traceformulas for groups of higher rank in order to analyze the problem of higher power moments of the zetafunction. The other was to exploit the finer group structure of the full modular group in order to get a more refined image of the zetafunction. Along the former line we could extract a fact that strongly suggested a possibility of an essential role playd by the SL (3, Z) traceformula 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 zetafunction. As for the research along the second line we report that we could establish a close relation between the Riemann zetafunction and Hecke congruence subgroups. It should be worth remarking that we found that Selbergs eigenvalue problem could be discussed in the frame of the theory of the power moments of the zetafunction. Less

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Research Products
(22 results)