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其他不同的问题。 最近，越来越明显的是，L 函数最自然地适合族，并且单个 L 函数的许多属性可以通过族来研究。 特别是，对 L 函数平均值的研究可以用来了解特殊值下 L 函数的大小（例如非零性和次凸性），这具有很大的算术意义。 此外，有证据表明，一些家庭与其他家庭有联系，这一属性是通过仔细评估某些平均值而发现的；例如，Motohashi 著名的精确公式将黎曼 zeta 函数的四阶矩与与全模群相关的 Hecke-Maass L 函数联系起来。 PI 打算使用解析数论技术来研究某些高阶矩问题，以深入了解更高阶的 L 函数。 此外，PI 寻求开发连接 L 函数族的新公式。黎曼 zeta 函数是 L 函数的基本示例。 正如黎曼猜想所预测的那样，素数（整数的基本粒子）的分布与黎曼 zeta 函数的属性交织在一起。 黎曼 zeta 函数本身只是复杂的 L 函数网络的一部分，这些函数本身以神秘的方式相互作用。 该项目旨在进一步理解黎曼 zeta 函数及其概括，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.