Statistical Questions in Number Theory and Arithmetic Geometry

数论和算术几何中的统计问题

• 批准号: 2001581
• 负责人: Paul Pollack
• 金额: $ 16.8万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001581&HistoricalAwards=false
• 关键词: Statistical; Questions; Number; Theory; Arithmetic

Number theory is a thriving area of mathematics research with deep connections to both computer science and digital security. The principal goal of this project is to deepen our understanding of certain ubiquitous number-theoretic objects (such as elliptic curves, power residues modulo prime numbers, and classical arithmetic functions such as Euler's phi-function), by studying them through a quantitative lens. One novel feature of this work is an approach to these problems that takes advantage of our understanding of the "anatomy of integers" describing the typical way a number breaks down into component parts (prime factors).Three main topics of investigation will be pursued: (1) A detailed study of the value-distribution of arithmetic functions, one particular problem being to pin down precise conditions under which a number has many factorizations. Here "factorizations" may be ordered or unordered, and we may wish to impose certain additional conditions on the parts. (2) The distribution of power residues. (3) The statistical distribution of torsion structures of elliptic curves. For (1) and (3), results from the "anatomy integers" will play a critical role. For (2), the PI plans to supplement the existing approaches to such problems - which have been mostly analytic, involving character sums - with algebraic approaches that make use of higher reciprocity laws.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.

数论是数学研究的一个蓬勃发展的领域，与计算机科学和数字安全有着密切的联系。这个项目的主要目标是加深我们对某些普遍存在的数论对象（如椭圆曲线，幂剩余模素数，和经典的算术函数，如欧拉的φ函数）的理解，通过定量透镜研究它们。这项工作的一个新特点是利用我们对“整数解剖学”的理解来解决这些问题，“整数解剖学”描述了数字分解为组成部分的典型方式（主要因素）。调查将围绕三个主要专题进行：（1）算术函数的值分布的详细研究，一个特殊的问题是确定一个数有许多因子分解的精确条件。这里的“因式分解”可以是有序的或无序的，并且我们可能希望对部分施加某些附加条件。(2)剩余功率的分布。(3)椭圆曲线挠结构的统计分布。对于（1）和（3），来自“解剖整数”的结果将起关键作用。对于（2），PI计划用代数方法来补充现有的解决这些问题的方法--这些方法大多是分析性的，涉及特征和--这些方法使用了更高的互惠律。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

The reciprocal sum of divisors of Mersenne numbers

梅森数因数的倒数和

• DOI: 10.4064/aa200602-11-9
• 发表时间: 2021
• 期刊: Acta Arithmetica
• 影响因子: 0.7
• 作者: Engberg, Zebediah;Pollack, Paul
• 通讯作者: Pollack, Paul

A Quick Route to Unique Factorization in Quadratic Orders

二次阶唯一因式分解的快速途径

• DOI: 10.1080/00029890.2021.1898875
• 发表时间: 2021
• 期刊: The American Mathematical Monthly
• 作者: Pollack, Paul;Snyder, Noah
• 通讯作者: Snyder, Noah

Intermediate prime factors in specified subsets

指定子集中的中间素因子

• DOI: 10.1007/s00605-023-01855-w
• 发表时间: 2023
• 期刊: Monatshefte für Mathematik
• 作者: McNew, Nathan;Pollack, Paul;Singha Roy, Akash
• 通讯作者: Singha Roy, Akash

Dirichlet, Sierpiński, and Benford

狄利克雷、西尔皮奥斯基和本福德

• DOI: 10.1016/j.jnt.2021.12.010
• 发表时间: 2022
• 期刊: Journal of Number Theory
• 影响因子: 0.7
• 作者: Pollack, Paul;Singha Roy, Akash
• 通讯作者: Singha Roy, Akash

Comparing multiplicative orders mod p , as p varies

比较乘法阶 mod p ，因为 p 变化

• 发表时间: 2021
• 作者: Matthew Just;P. Pollack
• 通讯作者: P. Pollack