Zeta Functions and Probability Theory

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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.