FRG: Averages of L-functions and Arithmetic Stratification

FRG：L 函数的平均值和算术分层

• 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

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.

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.

项目成果

Subconvexity in the inhomogeneous cubic Vinogradov system

非齐次三次维诺格拉多夫系统中的次凸性

• DOI: 10.1112/jlms.12698
• 发表时间: 2023
• 期刊: Journal of the London Mathematical Society
• 作者: Wooley, Trevor D.

A paucity problem for certain triples of diagonal equations

某些对角方程组的匮乏问题

• DOI: 10.1112/blms.12636
• 发表时间: 2022
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Brüdern, Jörg;Wooley, Trevor D.
• 通讯作者: Wooley, Trevor D.

Pairs Of Diagonal Quartic Forms: The Non-Singular Hasse Principle

对角四次形式对：非奇异哈斯原理

• DOI: 10.1093/qmath/haac019
• 发表时间: 2022
• 期刊: The Quarterly Journal of Mathematics
• 作者: Brüdern, Jörg;Wooley, Trevor D.
• 通讯作者: Wooley, Trevor D.

Optimal mean value estimates beyondVinogradov’s mean value theorem

超越维诺格拉多夫均值定理的最优均值估计

• DOI: 10.4064/aa200824-9-3
• 发表时间: 2021
• 期刊: Acta Arithmetica
• 影响因子: 0.7
• 作者: Brandes, Julia;Wooley, Trevor D.
• 通讯作者: Wooley, Trevor D.

Subconvexity in Inhomogeneous Vinogradov Systems

非齐次维诺格拉多夫系统中的次凸性

• DOI: 10.1093/qmath/haac027
• 发表时间: 2022
• 期刊: The Quarterly Journal of Mathematics
• 作者: Wooley, Trevor D.