Distributions in Multiplicative Number Theory

About This Grant

The PI will study distribution questions in L-functions and multiplicative functions, which are central subjects in multiplicative number theory. The first known example of L-functions is the Riemann zeta function. Classically, its zeros are connected to the distribution of prime numbers, which have applications to cryptography and modern security systems. There has been extensive research on the properties of L-functions, such as the distribution of their zeros and distribution of their values. However, many deep problems remain unsolved even though good conjectures have been formulated. The PI will explore various distributions in families of L-functions and provide insight into the structures of such families; the novel techniques developed will serve as powerful tools to shed light on other deep questions in the area. Another direction of the project is to explore ubiquitous statistical phenomenon known as Benford's law. This law first appeared as an observation about the first digits of the numbers in data sets. In particular, the leading digits do not exhibit uniform distribution as might be naively expected, but rather, the digit 1 appears the most, followed by 2, 3, and so on until 9. The goal is to give an answer in the context of multiplicative functions to the question "Is checking the first digit theoretically equivalent to checking many digits?" The award will provide opportunities for research training and collaboration for students and postdocs. The PI will also organize number theory seminars, AMS special sessions and continue engaging in outreach activities for middle school students and math circles. Many important conjectures on L-functions follow the Katz-Sarnak heuristic that the statistics of families of L-functions match the analogous statistics from classical compact groups of random matrices. In this direction, the PI will study the distribution of their values and zeros. In particular, the PI plans to study the nth centered moments of two families of GL(2) holomorphic L-functions. The aim is to provide the longest bandwidth in the literature as well as the application toward non-vanishing of higher order zeros of L-functions. Moreover, the PI will study various moments of L-functions (e.g. moments of twisted L-functions in a large orthogonal family, short moments of GL(4) x GL(2) Rankin Selberg L-functions) with applications toward critical line theorem, subconvexity and simultaneous non-vanishing of L-functions. These represent substantial progress in understanding open problems in the area, which are related to important conjectures such as the Generalized Riemann hypothesis (GRH) and the Katz-Sarnak philosophy. Moreover, the PI will investigate the Benford law phenomenon for multiplicative functions in connection with statistical universality and aim to deepen understanding of the structural interplay between harmonic analysis and multiplicative number theory. The methods employed in the proposal include harmonic analysis, development of new general Petersson's formula, random matrix theory, and Fourier analysis. 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.