On relations between homogeneous spaces and the Riemann zeta-function

齐次空间与黎曼 zeta 函数的关系

基本信息

批准号：
05804004
负责人：
MOTOHASHI Yoichi
金额：
$ 1.09万
依托单位：
Nihon University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1993
资助国家：
日本
起止时间：
1993 至 1994
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-05804004/
关键词：
Zeta-Function Homogeneous Spaces Distribution of Primes 保型形式整数論

项目摘要

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

黎曼 zeta 函数与素数分布之间的关系传统上是考虑到它们与黎曼假设的直接相互作用而进行讨论的，因此研究的重点是 zeta 函数的定性方面。然而，我们的研究是将我们的注意力转移到这个基本函数的定量方面，例如，众所周知的事实表明，就短间隔内的素数分布而言，这一论点是可行的。关注黎曼假设可能会被 zeta 函数的某种定量性质所取代，后者是 zeta 函数沿临界线的值的矩问题，直到我们最近成功建立之前，一直用纯经典的方法进行讨论。它与双曲拉普拉斯光谱分辨率的关系，特别展示了一种新观点的可能性，即黎曼 ze … 更多 ta 函数可能被视为 Hecke L 函数（马斯波）的生成器。换句话说，zeta 函数的定量性质包含一些代表双曲平面结构的波分量，为此，我们采用了两种方法。借助高阶群的迹公式理论来分析zeta函数的更高幂矩问题，另一个是利用全模群的更精细的群结构以获得更精细的群结构。精致的形象沿着前一条线，我们可以得出一个事实，该事实强烈表明 SL (3, Z) 迹公式在六次幂矩问题的理论中可能发挥重要作用。强调我们还发现，与预期相反，8 次幂矩问题可以简化为 SL(2, Z) 理论，这一发现似乎表明了 zeta 函数幂矩理论的新前景。至于沿途的研究第二行我们报告我们可以在黎曼 zeta 函数和 Hecke 同余子群之间建立密切的关系值得一提的是我们发现 Selberg 的特征值问题可以在幂矩理论的框架中讨论。小于 zeta 函数。