Mean values f L-functions

-函数在数论中起着统一的作用，因为它们连接了许多不同的问题。最近，越来越明显的是，L函数最自然地适合于家庭，并且单个L函数的许多性质可以通过家庭来研究。特别地，对L函数均值的研究可以用来理解L函数在特定值下的大小(如非零性和次凸性)，这具有很大的算术意义。此外，有证据表明一些家族与其他家族有联系，这一性质是通过仔细评估某些平均值而发现的；例如，Motohashi著名的精确公式将Riemann Zeta函数的四阶矩与与完全模群相关的Hecke-Maass L函数联系起来。PI打算用解析数论中的技巧来研究某些高阶矩问题，以获得对高次L函数的洞察。此外，PI还寻求建立连接L函数族的新公式。Riemann Zeta函数是L函数的基本例子。素数的分布，即整数的基本粒子，与黎曼Zeta函数的性质交织在一起，正如黎曼假设所预测的那样。黎曼·泽塔函数本身只是L复杂网络的一部分，这些函数本身以神秘的方式相互作用。这个项目旨在加深对黎曼-泽塔函数及其推广的理解，特别是通过寻找不同类型的L函数之间的新联系。

-functions play a unifying role in number theory, as they connect many otherwise different problems. Recently it has become increasingly apparent that L-functions most naturally fit into families, and that many properties of an individual L-function can be studied through the family. In particular, the study of mean values of L-functions can be used to understand the sizes (such as nonvanishing and subconvexity) of L-functions at special values, which have great arithmetical interest. Furthermore, there is evidence that some families are connected to other families, a property that has been detected through careful evaluation of certain mean values; for example, Motohashi's celebrated exact formula connecting the fourth moment of the Riemann zeta function to the Hecke-Maass L-functions associated to the full modular group. The PI intends to investigate certain higher moment problems using techniques from analytic number theory to gain insight into higher degree L-functions. Furthermore, the PI seeks to develop new formulas connecting families of L-functions.The Riemann zeta function is the basic example of an L-function. The distribution of the prime numbers, the fundamental particles of the integers, are intertwined with properties of the Riemann zeta function, as predicted by the Riemann Hypothesis. The Riemann zeta function itself is just one part of a complex web of L-functions that themselves interact in mysterious ways. This project aims to further the understanding of the Riemann zeta function and its generalizations, especially by finding new links between different kinds of L-functions.