Zeta Functions and Probability Theory

中心极限定理在数学和其他领域中无处不在。 它在200年内的演变代表了数学编年史上重要的一章。在数论，转折点是1940年定理的鄂尔多斯和卡茨承认，经典定理的哈代和拉马努金关于一些素因子的随机整数是真正的一个定理在概率论的启发中心极限定理。 不久之后，库比留斯创造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推论，而是由中心极限定理提出的，因为相关的“随机变量”不一定像定理所要求的那样是独立的。 但是，它们是“近似独立的”。 而这正是关键所在。 在每一种应用情况下，特别是在数论中，缺乏独立性是巧妙的方便条件。 这项研究计划的一个主要部分是研究如何使用中心极限定理和概率论的其他相关定理（如重对数定律）的隐喻来扩大我们对zeta函数的理解，特别是当它与著名的黎曼假设有关时。 更具体地说，隐喻可以用来做出预测和合理的假设，人们可以使用数论和概率的工具来接近这些假设。 这是这项建议的中心主题。 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。 使用l进表示理论结合切博塔列夫密度定理，我们可以证明一个类似的埃尔多斯-卡茨定理假设一个“准”广义黎曼假设。 这项研究提案的一部分旨在消除这一未经证实的假设。 在拟议的研究中，我们计划调查的正常数量的素因子的拉马努金tau函数在移位的素数。 这是我给我现在的博士生Arpita Kar的研究的原始问题。 这项工作还表明，一个类似的问题移位的素数的经典欧拉Φ函数这是从来没有考虑过，可以解决基本上使用塞尔伯格筛，Bjueri-Vinogradov定理和定理从概率论。 我预计这项研究将在未来两年内完成。 在此之后，人们考虑联合分布，这需要最近的进展理论的l-adic表示附加到两个本征形。 这些拟议工作将包括拟议研究的头三年。 在第四年和第五年，我希望指导4名研究生和2名博士后。 他们的研究将涉及在两个方向上扩展这项工作。 一个将是考虑几个特征形和另一个将是研究这些算术中心极限定理的误差项。

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.