The Distribution of Zeros and Values of the Riemann Zeta-Function and L-Functions

这个项目的目的是研究黎曼Zeta函数和其他L函数的行为的基本问题。作者将用新的方法研究Zeta函数在临界线右侧垂直线附近导数的零点分布。实证研究表明，这些零点的分布函数具有具有两个局部极大值的特点。也许这里提出的理论和数值研究可以解释这一现象。第二个项目是证明H.L.Montgomery和提出者最近关于Zeta函数在其临界点的大值的猜想。这些猜想是在他们研究Zeta函数通过这些点的水平曲线的几何时产生的。作者还将研究L函数的a点在临界线上的分布以及简单a点在临界线右侧的比例。这里的一个目标是通过证明在任何情况下，最多只有无穷小的比例，来支持Selberg的猜想，即在临界线上只有有限多的a点。其他问题是确定Zeta函数实部和虚部零点的对相关函数，并确定当一个人接近临界线时它是如何变化的；使用提出的Zeta函数的S混合公式和随机矩阵理论建模来计算Zeta函数导数的离散矩；确定Zeta函数零点的备选假设和备选方案之间的联系。最后，通过将这些结果推广到很长的积，显著地改进了作者最近关于有限欧拉积的矩的工作。提案中描述的项目都与解析数论中的基本问题有关，即Selberg类中的Riemann Zeta函数和其他L函数的解析和几何性质。这些进展将推动Zeta函数理论和其他L函数理论在传统和新方向上的发展。由于大多数问题与调和分析、概率论和复函数理论有关，所提出的一些方法是新的，并且很可能应用到其他领域。与随机矩阵理论中的问题也有直接的联系，如随机矩阵的特征多项式的导数的零点分布。

The goal of this project is to investigate fundamental questions about the behavior of the Riemann zeta-function and other L-functions. The proposer will study the distribution of zeros of the derivative of the zeta-function near vertical lines to the right of the critical line by new methods. Empirical studies suggest that the distribution function of these zeros has the peculiar feature of possessing two local maxima. It may be that the theoretical and numerical studies proposed here will explain this phenomenon. A second project is to prove the recent conjectures of H. L. Montgomery and the proposer concerning large values of the zeta-function at its critical points. These conjectures arose in their study of the geometry of the level curves of the zeta-function through these points. The proposer will also investigate the distribution of a-points of L-functions on the critical line and the proportion of simple a-points to the right of it. A goal here is to support Selberg's conjecture that there are only finitely many a-points on the critical line by showing that, in any case, at most an infinitesimal proportion are. Other problems are to determine the pair correlation function of the zeros of the real and imaginary parts of the zeta-function and to determine how it changes as one approaches the critical line; to calculate discrete moments of the derivative of the zeta-function using the proposer?s hybrid formula for the zeta-function and random matrix theory modeling; to determine the connection between the Alternative Hypothesis for the zeros of the zeta-function and an ?alternative? twin prime conjecture; and, finally, to improve significantly the proposer's recent work on moments of finite Euler products by extending these results to very long products.The projects described in the proposal are all concerned with fundamental issues in analytic number theory, namely, the analytic and geometric properties of the Riemann zeta-function and other L-functions in the Selberg class. Progress will advance the development of the theory of the zeta-function and these other L-functions in both traditional and new directions. Some of the proposed methods are new and may well have applications to other areas since most of the problems have connections with harmonic analysis, probability theory, and complex function theory. There are also direct connections with questions in random matrix theory such as the distribution of zeros of the derivative of characteristic polynomials of random matrices.