FRG: Averages of L-functions and Arithmetic Stratification

FRG：L 函数的平均值和算术分层

基本信息

批准号：
1854398
负责人：
John Conrey
金额：
$ 120万
依托单位：
American Institute of Mathematics
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854398&HistoricalAwards=false
关键词：
FRG Averages functions Arithmetic Stratification

项目摘要

Some of the most difficult challenges in all of mathematics, such as the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture, are naturally phrased in terms of L-functions. These functions encode information such as how many primes there are up to a given magnitude, or the frequency of rational number solutions to certain equations, or the distribution of special points on a surface, all of which are important in number theory. L-functions are often studied in collections called families. In this project we will use a new approach called "stratification" to study the distribution of the values of L-functions in families.In recent years researchers have found very precise conjectures about the statistics of values and zeros of the Riemann zeta function and other families of L-functions. For low order moments these conjectures follow from precise knowledge or conjectures about correlations of generalized divisor functions. But for higher moments this linkage has been missing. The main goal of this project is to complete this picture and prove that the moment conjectures for families of L-functions follow from knowledge of divisor correlations, which is equivalent to counting points in specified regions of certain varieties. We will also investigate the same scenario but for averages of ratios of L-functions in families with the divisor correlations replaced by the general Hardy-Littlewood conjectures about prime tuples. For the Riemann zeta-function the project will follow the method outlined in recent work by two of the PIs. For other families of L-functions another innovation is required. This project will be informed by Manin's ideas for counting rational points on varieties by identifying the stratified subvarieties that play a role and counting the points on these. We will also investigate the exponential sums that naturally arise in counting points on these subvarieties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在所有数学中，一些最困难的挑战，例如Riemann假设以及Birch和Swinnerton-Dyer猜想，自然而然地用L功能来表达。这些函数编码信息，例如有多少次数到给定的幅度，或某些方程式的有理数解决方案的频率，或在表面上的特殊点的分布，所有这些方程式在数字理论中都很重要。经常在称为家庭的收藏中研究L功能。在这个项目中，我们将使用一种称为“分层”的新方法来研究家庭中L功能值的分布。近年来，研究人员发现了关于Riemann Zeta功能和其他L函数家族的价值和零元素的统计数据的非常精确的猜想。在低阶时刻，这些猜想源于关于广义除数函数相关的精确知识或猜想。但是在更高的时刻，这种联系一直丢失。该项目的主要目的是完成这张图片，并证明L功能家庭的猜想源于对除数相关性的知识，这等同于某些品种指定区域中的计数点。我们还将调查相同的情况，但是对于由一般的Hardy-Little Wood猜想代替了关于主要元素的一般性的家庭的L功能比率的平均值。对于Riemann Zeta功能，该项目将遵循两个PI的最新工作中概述的方法。对于其他l功能家庭，需要另一个创新。 Manin的想法将通过确定发挥作用并计算这些要点的分层次变量来了解该项目。我们还将调查这些亚各个次数计数点自然出现的指数款项。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来获得支持的。