L-函数、大值特征和及相关问题研究

项目名称： L-函数、大值特征和及相关问题研究

项目编号： No.11471258

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 徐哲峰

作者单位： 西北大学

项目金额： 75万元

中文摘要： L-函数与特征和的估计问题都是解析数论研究的中心问题，许多数论问题都与它们密切相关。近年来，由于新方法的引入，在这两方面的研究工作都取得了突破性的进展。A. Granville与K. Soundararajan通过对大值特征和取值规律本质的分析，获得了一系列有关特征和的重要估计，并对奇数阶原特征的情形很大程度地改进了Polya-Vinogradov上界。本项目拟通过对不同特征和与相应L-函数关系的深入研究，并结合共鸣方法与解析技巧研究大值特征和在不同区间或不同数集上的上下界估计以及高次均值估计问题以及有限域、椭圆曲线、正规群、多项式环以及某些特殊数集上特征和的上下界估计或计算问题，并利用随机矩阵理论和解析方法研究一些L-函数在关键区域的上下界估计与均值分布性质以及这些性质在特征和、Gauss和、Dedekind和与整数及其逆的分布等数论问题中的应用。

中文关键词： L-函数；大值特征和；均值；上下界估计

英文摘要： The estimation for L-function and character sum are all the central problem in analytic number theory. They have colse relations with many number theory problems and very important applications. Recently, due to the introduction of new methods, number theorists have made breakthrough in this two problems. By analyzing the essence of value distribution of large character sums, A. Granville and K. Soundararajan have obtained some significant estimation for character sums and greatly improved the famous Polya-Vinogradov upper bound for primitive characters of odd order. By researching the relations between various character sums and their corresponding L-functions,combining resonance method and analytic technique, we will study the upper or lower bound esitimation and mean value distribution of large character sums over different intervals or number sets, upper or lower bound estimations or computations of character sums over finite fields, elliptic curves, normal groups, polynomial rings and some special number sets. Combining random matrix theory and analytic mehtods, we also study the lower or upper bound estimates or distribution properties of mean values of some L-functions on crucial region, and to study the application of these mean values of L-functions in character sum, Gauss sum, Dedekind sum and some number theory problem such as distribution of integer and its inverse.

英文关键词： L-function;Large character sums;Mean value;Lower or upper bound estamate