Dirichlet L 函数零点的若干性质与素数间隙的遍历性质研究

Properties of Dirichlet L-functions and distribution of primes are very important in research of number theory，and have a very close relationship. This project aims at the study of distribution of simple zeros, gaps between consecutive zeros of the Dirichlet L-functions and distribution of gaps among primes. It is believed that all zeros of the Riemann zeta-function and Dirichlet L-functions are simple, which is still open now. On this problem, the project will choose a different entry point than before. By studying the number of additional zeros caused by multiplicity and combining research of gaps between consecutive zeros, we may partition the critical area into some sub areas and study every sub area independently. This way will expand research contents of simple zeros, and be much more likely to impove efficacy of the study on the problem. In the process of the project, we will also further study the distribution of gaps among primes by conbining the theory of the Riemann zeta-function, Dirichlet L-function and ergodic theory, to explore ergodic properties of distribution of gaps among primes. To this aim, we have to make a breakthrough at some new idea and creative technique.

Dirichlet L函数零点的性质和素数分布问题是数论中非常重要的研究对象，而且关系紧密。本项目将研究Dirichlet L函数的单零点分布、零点的纵向分布(间隙问题)以及素数间隙的分布。人们一直相信Riemann zeta函数和Dirichlet L函数的所有非显然零点都是单的，但至今无法证明。在该问题上，本项目将选取新的切入点，从零点多重性所产生的额外零点出发，结合零点间隙问题的研究，把主要区域划分成很多个小区域，再对每个小区域独立研究。这一措施在单零点数量的估计上丰富了研究内容和范围，并在申请人正在进行的初步工作中明显提高了估计的有效性。本项目还将结合遍历论和Riemann zeta函数、Dirichlet L函数的相关性质对素数间隙的分布情况进行深入的研究，揭示素数间隙分布的遍历性质，探索素数的分布规律，力争在理论和方法上取得突破，实现拟定的研究目标。

Dirichlet L 函数的零点性质和素数分布情况是数论中重要的研究对象，且关系紧密。在本课题的资助下我们研究了Dirichlet L函数的单零点问题和素数差集的大小问题，共发表标注资助的SCI论文3篇。主要取得了如下研究成果：a) 证明了Riemann zeta函数的非显然零点中超过66.036%是互异的；b) 证明了所有Dirichlet L函数的非平凡零点中超过60.261%是单零点且超过80.13%是互异的；c) 能够被表示为无穷对素数差的偶数集是Delta_r*集。这些研究成果推进了之前的结果并加深了人们对相关问题的认识。