Euler Product Models of L-Functions and the Distribution of Zeros and Primes

作者最近构造了一个函数参数族，其成员近似黎曼ζ函数或其他l -函数，其结构包含了这些函数的基本特征，如它们的欧拉积和函数方程。族中的每个函数都满足一个黎曼假设，并有有限多个可能的例外。此外，当参数不太大时，函数的零点数与zeta或l函数的零点数大致相同，零点都是简单的，连续的零点相互排斥。因此，我们可以把它们看作是函数或l函数的模型。事实上，如果黎曼假设对l函数成立，随着参数的增加，对应族中的函数的零点趋向于l函数的零点，并且在临界线上的零点之间，函数趋向于l函数的两倍。该项目的主要目标是进一步研究新函数，以深入了解黎曼ζ函数和l函数的行为以及它们的零和素数之间的联系。一个项目是调查模型在靠近临界线的过渡区域近似l函数的程度。另一种是研究一个特别重要的参数范围的近似值。在几个项目中，提案人将使用这些模型来试图理解迄今为止无法用其他方法解决的问题，例如ζ函数导数的零点是如何分布的。目前的研究路线使有限欧拉积在解析数论中比以前发挥了更突出的作用，因此另一个目标是重新研究这些积，特别是它们的矩。从传统的观点来看，这是相当困难的，但提议者最近的工作带来了一种新的方法。第二个独立的项目旨在进一步加深我们对l函数的素数和零的分布的理解。一是探索高斯酉系综假设、素数分布和ζ函数均值之间的联系。另一种方法是计算ζ函数零点的离散数方差和其他统计量。第三个项目是开发长狄利克雷多项式的离散均值公式，并使用这些公式来估计ζ函数零点之间的间隙大小。所提出的项目都涉及解析数论中的基本问题：素数的分布，l函数的零的分布，以及l函数的一般行为。素数是算术的基本组成部分，因此也是数学的基本组成部分，所以了解它们的性质是非常重要的。黎曼ζ函数和其他l函数中都包含了这些性质，这也使得它们成为重要的研究对象。提出的大多数项目都集中在研究作者最近构造的函数上，以建模l函数并捕获它们的基本特征。这些简单函数的结构清楚地说明了它们的行为，从而提供了对实际l函数行为的洞察。其余的项目更直接地研究素数、l函数及其联系。

The proposer recently constructed one parameter families of functions whose members approximate the Riemann zeta-function or other L-functions, and whose structure incorporates the fundamental features of these functions, such as their Euler products and functional equations. Each function in a family satisfies a Riemann hypothesis with finitely many possible exceptions. Moreover, when the parameter is not too large, the functions have approximately the same number of zeros as the zeta or L-function, the zeros are all simple, and consecutive zeros repel. One may therefore regard them as models of the zeta-function or L-function. In fact, if the Riemann hypothesis holds for an L-function, the zeros of functions in the corresponding family tend to those of the L-function as the parameter increases, and between zeros on the critical line the functions tend to twice the L-function. The main goal of this project is to investigate the new functions further in order to gain insight into the behavior of the Riemannn zeta-function and L-functions and into connections between their zeros and the prime numbers. One project is to investigate how well the models approximate L-functions in a transitional region close to the critical line. Another is to study the approximations for a particularly important range of the parameter. In several projects the proposer will use the models to try to understand problems that have so far resisted other treatments, such as how the zeros of the derivative of the zeta-function are distributed. The present line of inquiry gives finite Euler products a more prominent role than previously in analytic number theory, so another goal is to study such products anew, particularly their moments. This is quite difficult from the traditional point of view, but the prposer's recent work leads to a new approach. A second and separate set of projects aims at furthering our understanding of the distribution of primes and zeros of L-functions. One is to explore connections between the Gaussian Unitary Ensemble hypothesis, the distribution of primes, and mean values of the zeta-function. Another is to calculate the discrete number variance and other statistics for the zeros of the zeta-function. A third project is to develop discrete mean values formulas for long Dirichlet polynomials and use these to estimate the size of gaps between zeros of the zeta-function.The projects proposed all address fundamental problems in analytic number theory: the distribution of prime numbers, the distribution of zeros of L-functions, and the general behavior of L-functions. Prime numbers are the ultimate building blocks of arithmetic, and therefore much of mathematics, so understanding their properties is of basic importance. Many of these properties are encoded in the Riemann zeta-function and other L-functions, and that makes these important objects of study as well. Most of the projects proposed center on the investigation of functions recently constructed by the proposer to model L-functions and capture their basic features. The structure of these simpler functions makes it clear why they behave as they do, thereby providing insight into the behavior of the actual L-functions. The remaining projects study the primes, L-functions, and their connections more directly.