Mathematical Sciences: Analytic Theory of L-functions

l -函数理论是数论问题研究的核心，特别是在解析数论领域。对于大多数应用，相关问题通常涉及l函数的解析延拓，它们的函数方程，它们在垂直条中的增长估计，以及它们的零分布的知识。本项目涉及l -函数理论中非常一般的问题（Dirichlet级数与函数方程和欧拉乘积的分类及其与自同构形式的联系）和特殊问题（特定l -函数的中值定理，以及它们的零的分布）。本研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。

The theory of L-functions is central in the study of number theoretical questions, especially in the realm of analytic number theory. For most applications, the relevant questions usually involve the analytic continuation of the L-functions, their functional equations, estimates for their growth in vertical strips, and knowledge of the distribution of their zeros. This project deals with both very general questions (the classification of Dirichlet series with functional equations and Euler products and their connections with automorphic forms) and particular questions (mean-value theorems for particular L-functions, and the distribution of their zeros) in the theory of L-functions. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.