Zeta Functions and Probability Theory

Zeta 函数和概率论

• 批准号: RGPIN-2020-03927
• 负责人: Murty, Ram
• 金额: $ 3.13万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716143
• 关键词: Zeta; Functions; Probability; Theory

The central limit theorem has an ubiquitous presence in mathematics and beyond. Its evolution over a span of 200 years represents a significant chapter in the annals of mathematics. In number theory, the turning point was the 1940 theorem of Erdos and Kac which recognized that the classical theorem of Hardy and Ramanujan concerning the number of prime factors of a random integer was really a theorem in probability theory inspired by the central limit theorem. Shortly after, Kubilius created a new branch of mathematics called probabilistic number theory. These theorems and results are not a consequence of the central limit theorem but are suggested by it because the relevant "random variables" are not necessarily independent as required by the theorem. However, they are "approximately independent". And this is precisely the point. In each case of application especially in number theory, the lack of independence is finessed by convenient conditions. A major part of this research proposal is to study how the metaphor of the central limit theorem and other related theorems of probability theory (such as the law of the iterated logarithm) can be used to enlarge our understanding of zeta functions, especially as it relates to the celebrated Riemann hypothesis. More specifically, the metaphor can be used to make predictions and reasonable conjectures and one can approach these conjectures using tools from both number theory and probability. This is the central theme of this proposal. In 1984, Kumar Murty and I initiated the study of the normal number of prime factors of Fourier coefficients of modular forms. Using the theory of l-adic representations combined with the Chebotarev density theorem, we could prove an analog of the Erdos-Kac theorem assuming a "quasi" generalized Riemann hypothesis. Part of this research proposal is aimed at eliminating this unproved hypothesis. In proposed research, we plan to investigate the normal number of prime factors of the Ramanujan tau-function at shifts of prime numbers. This was the original problem for research that I had given my current doctoral student, Arpita Kar. This work also suggested that an analogous question for shifts of primes of the classical Euler phi-functions which was never considered before, can be solved using essentially the Selberg sieve, the Bombieri-Vinogradov theorem and theorems from probability theory. I expect this research will be completed in the next two years. After this, one considers joint distributions and this needs recent advances in the theory of l-adic representations attached to two eigenforms. These proposed works will comprise the first three years of proposed research. In years 4 and 5, I expect to supervise 4 graduate students and 2 postdocs. Their research will involve extending this work in two directions. One will be to consider several eigenforms and another will be to study error terms in these arithmetical central limit theorems.

中心极限定理在数学和其他领域中无处不在。它在200年的时间里的演变是数学史上一个重要的篇章。在数论中，转折点是1940年Erdos和Kac的定理，该定理认识到Hardy和Ramanujan关于随机整数的素因子数的经典定理实际上是受中心极限定理启发的概率论中的一个定理。不久之后，库比略创立了一个新的数学分支，称为概率数论。这些定理和结果不是中心极限定理的结果，而是由中心极限定理提出的，因为相关的“随机变量”不一定像该定理所要求的那样是独立的。然而，它们“几乎是独立的”。这正是问题的关键所在。在每一种应用中，特别是在数论中，独立性的缺乏都是由方便的条件巧妙地解决的。这项研究建议的一个主要部分是研究如何利用中心极限定理和其他概率论相关定理(如重对数律)的隐喻来扩大我们对Zeta函数的理解，特别是当它与著名的Riemann假设有关时。更具体地说，隐喻可以用来做出预测和合理的猜测，人们可以使用数论和概率论的工具来处理这些猜测。这是这项提议的中心主题。 1984年，Kumar Murty和我开始研究模形式的傅里叶系数的素因子的正规数。利用L表示理论与契波塔列夫密度定理相结合，我们可以在“准”广义黎曼假设下证明类似的Erdos-Kac定理。这项研究提案的一部分旨在消除这一未经证实的假设。 在所提出的研究中，我们计划研究Ramanujan tau-函数在素数移位时素数因子的正规数。这是我给我现在的博士生阿皮塔·卡尔的研究的原始问题。这项工作还表明，利用Selberg筛子、Bombieri-Vinogradov定理和概率论中的定理，可以解决以前从未考虑过的经典Euler Phi函数的素数移位的类似问题。我预计这项研究将在未来两年内完成。在此之后，考虑联合分布，这需要在附加于两个本征形式的L-进表示理论方面的最新进展。这些拟议工程将包括拟议研究的首三年。 在第四年和第五年，我希望指导4名研究生和2名博士后。他们的研究将涉及将这项工作扩展到两个方向。一种是考虑几种特征形式，另一种是研究这些算术中心极限定理中的误差项。