Zeros and Moments of L-Functions

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This award is in the area of analytic number theory, focusing on the study of L-functions, which play a central role in the field. L-functions are generalizations of the Riemann zeta function, which encodes important information about the distribution of prime numbers. More generally, L-functions package information about important arithmetic objects (such as the rank of an elliptic curve or the class number) which are of great interest. This project aims to develop new tools in the study of L-functions, particularly by focusing on their zeros and their central values, with the hope of extracting arithmetic information. The investigator will advise PhD students and provide mentorship to students coming from groups that are underrepresented in mathematics.The award will study moments and ratios of L-functions in families, both in the number field and in the function field setting. The final goal is that of understanding the mechanism through which lower-order terms in the moment asymptotics work and the structure of each family. Another theme of the project is studying zeros of L-functions in families, focusing on obtaining vanishing and non-vanishing results. The methods employed will be analytic number theory techniques, including use of functional equations, summation formulas, exponential sums, sieve theory ideas and random matrix theory inspired insights.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.