代数迹函数及其在现代解析数论中的应用

This program is devoted to the study of analytic theory of algebraic trace function, as well as their applications to modern analytic number theory. In particular, we study the average behaviors of algebraic trace functions, mod p or q, over some specialized sequences (such as smooth numbers, arithmetic progressions), as well as their applications in the equidistribution of arithmetic functions in arithmetic progressions, Sato-Tate conjecture of Kloosterman sums and so on. Furthermore, the analytic theory of composite trace functions will also be developed, and the method of arithmetic exponent pairs and the q-analogue of van der Corput method will be among the most important tasks, from which some important applications, including bilinear forms of trace functions, averages over primes, moments of automorphic L-functions, will be derived. In addition, large sieve inequalities for general families of algebraic trace functions will be established, keeping in mind applications to the equidistributions of many arithmetic and geometric objects...Algebraic trace functions contribute as one of the most important objects in the current flow of modern number theory and arithmetic algebraic geometry, and their analytic theory, rooted in arithmetic geometry, provide powerful tools for various problems in modern analytic number theory.

本项目主要研究代数迹函数的解析理论及其在现代解析数论中的具体应用. 具体包括素域上的代数迹函数在特殊序列(如脆数、等差数列等)中的均值, 模q的复合迹函数相应的解析性质, 并将其应用于算术函数在算术级数中的平均分布、Kloosterman和的Sato-Tate猜想等问题中; 进一步发展算术型指数对理论以及有限的van der Corput方法, 并着手研究二维的情形, 从而对代数迹函数的双线性型给出更优的估计, 并将其应用于迹函数的素变量均值、L函数的均值等重要问题之中. 此外, 建立一般代数迹函数族的大筛法型不等式, 将其应用于若干算术与几何对象的平均分布中, 并证明相应的Bombieri-Vinogradov型定理. ..代数迹函数是现代数论与算术代数几何的重要研究对象, 它的解析理论为解析数论的发展提供了强有力的工具, 值得进行深入、系统的研究.

代数迹函数是现代数论与算术代数几何的重要研究对象, 它的解析理论为现代解析数论的发展提供了强有力的工具, 值得进行深入系统的研究...本项目进一步完善了代数迹函数的算术型指数对理论, 实质刻画了Kloosterman和的模结构, 从而从“殆素数”角度否定回答了Nicholas Katz的问题. 此外, 还利用代数迹函数的解析理论研究了素数分布、自守形式及Pell方程中的若干问题. 从殆素数值、最大素因子的角度研究了二次多项式的Schinzel猜想. 特别的, 证明了存在无穷多个素数p, 使得p^2+2至多有4个素因子, 从而改进了Richert 50年前的结果. 给出了Hooley猜想中关于Pell方程基本解的计数函数的下界不等式, 改进了Fouvry和Bourgain等人的工作...我们相信, 在未来的工作中能够证明代数迹函数的更多解析性质, 并将其应用于更多解析数论问题的研究.