Mean values f L-functions

函数在数论中起着统一的作用，因为它们把许多不同的问题联系在一起。最近，越来越明显的是，l -函数最自然地适合于族，并且单个l -函数的许多性质可以通过族来研究。特别是，研究l函数的均值可以用来理解l函数在特殊值处的大小（如非消失性和次凸性），这具有很大的算术意义。此外，有证据表明，一些家庭与其他家庭有联系，这是通过对某些平均值的仔细评估发现的。例如，Motohashi著名的精确公式将Riemann zeta函数的第四矩与与全模群相关的hecke - mass l函数联系起来。PI打算利用解析数论的技术来研究某些高矩问题，以深入了解高次l函数。此外，PI还寻求开发连接l函数族的新公式。黎曼函数是l函数的基本例子。正如黎曼假设所预测的那样，素数（整数的基本粒子）的分布与黎曼ζ函数的性质交织在一起。黎曼ζ函数本身只是l函数复杂网络的一部分，这些l函数以神秘的方式相互作用。本项目旨在进一步了解黎曼ζ函数及其推广，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.