关于黎曼假设的研究

Riemann hypothesis is one of the most important topics in mathematics. Not only it is related to the distribution of primes, but also represents some general rules of our physical world. In quantum physics, Hilbert spaces and operators are basic tools to describe the physical world. Based on Hilbert spaces, the research has three approaches as follows...1. "Zeros of Riemann ζ function and spectrum of the operator". New Hilbert space and operator will be constructed such that the image parts of zeros of Riemann ζ function will belong to the spectrum of the operator. Then Riemann hypothesis will be reduced to the self-adjointness of the operator...2. "Closure equivalent form of Riemann hypothesis". New Hilbert space and family of functions will be constructed, and Riemann hypothesis will be reduced to the denseness of a family of functions in the space...3. "Positivity of Riemann hypothesis". Riemann hypothesis will be studied through the properties at s=1 of the function ζ(s)...This program aims at finding new methods for the research of Riemann hypothesis.

黎曼假设是数学中最重要的课题之一，不仅与素数分布直接相关，更代表了物理世界中的某种普遍规律. Hilbert空间和算子是量子物理描述世界的基本方式，本项目将以Hilbert空间为支点，从下述三条途径出发开展研究. 一、“黎曼ζ函数零点与算子的谱”：构造Hilbert空间和其上的算子，使得黎曼ζ函数的零点虚部落在该算子的谱中，从而将黎曼假设归结为算子的自伴性；二、“黎曼假设的稠密性等价命题”：构造Hilbert空间和其中的函数族，将黎曼假设归结为函数族在空间中的稠密性；三、“黎曼假设中的正定性”：将黎曼假设归结为函数ζ(s)在点s=1处的性质进行研究. 项目将致力于为黎曼假设的研究开启全新的篇章.

黎曼假设是数学中最重要的课题之一，不仅与素数分布直接相关，更契合物理世界中的一些普遍规律。本项目从Hilbert空间和算子出发，对黎曼假设展开研究。主要成果包含以下几个方面：黎曼ζ函数零点与自伴的乘性微分算子的谱一一对应；自然数左正则表示生成的各类算子代数与自然数的乘法结构的对应；黎曼假设在加权Bergman空间上的稠密性等价命题以及不变子空间等价命题；建立非交换的Thompson半群上的初步算术理论；关于一些和积问题的改进等等。本项目尝试了一些研究黎曼假设的新途径，展开了比较新颖的视角，并提出了一些新问题。

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