Analytic Theory of L-functions

L函数是数论的基本函数，在素数的研究、整数方程的求解、算术序列分布的一致性等方面都有重要的作用。第一个L函数是黎曼Zeta函数。它的零点的位置是黎曼假设的主题，该假设被广泛认为是所有数学中最重要的悬而未决的问题。在这个项目中，PI将研究黎曼假说的各种方法。此外，他还将研究黎曼Zeta函数和L函数族的一些非常特殊的统计性质，以便更全面地理解这些神秘的函数。Pi和他的合作者打算证明Dirichlet L函数的大多数零点都在临界线上。他们还将开发一个通用工具，渐近大筛子，来解决涉及模数小于给定参数的所有本原Dirichlet特征标的平均值的问题。他们将利用这一点来研究狄里克莱特L函数的零点间距。另一个项目是确定(猜想)Riemann Zeta函数的连续零点之间的间隔分布的算术部分。最后，PI想要证明他猜想的Vasyunn和的互易公式，并研究它在Nyman-Beurling方法中对Riemann假设的相关性。

L-functions are the basic functions of number theory and are fundamental in the study of prime numbers, solutions to equations in whole numbers, and the uniformity of the distribution of arithmetic sequences. The first L-function is the Riemann zeta-function. The location of its zeros is the subject of the Riemann Hypothesis which is widely regarded as the most important unsolved problem in all of mathmatics. In this project the PI will study various approaches to the Riemann Hypothesis. In addition, he will investigate some very specific statistical properties of the Riemann zeta-function and of families of L-functions in order to more fully understand thes mysterious functions.The PI and his collaborators intend to prove that most of the zeros of Dirichlet L-functions are on the critical line. They also will to develop a general tool, the asymptotic large sieve, to address problems involving averages over all primitive Dirichlet characters of modulus less than a given parameter. They will use this to investigate the spacings between zeros of Dirichlet L-functions. Another project is to determine (conjecturally) the arithemetic part of the distribution of spacings between consecutive zeros of the Riemann zeta-function.Finally, the PI would like to prove the reciprocity formula that he conjectured for Vasyunn sums, and to investigate its relevance in the Nyman-Beurling approach to the Riemann Hypothesis.