The Distribution of Zeros of L-Functions and Related Questions

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L-functions have played a pivotal role in the modern development of number theory and they can be used to study a wide variety of problems. The tools used to study L-functions draw from many branches of mathematics including analysis, algebra, algebraic geometry, representation theory, and mathematical physics while number theory has important applications outside of mathematics to fields such as theoretical computer science and cryptography. Many of the topics investigated in this project concern the distribution of zeros of L-functions and related topics in arithmetic. This relationship is central to two of the seven Millennium Prize Problems, the Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture. The investigator will continue training and mentoring graduate students in this research area, and this project will provide research training opportunities for them. One of the problems in this project will investigate a thin family of Dirichlet L-functions which experimentally has significant and previously undetected bias in distribution of the gaps between their zeros. This bias seems to have an arithmetic explanation that corresponds to the non-vanishing of a certain Gauss type sum connected to moments of these L-functions. Other problems study the distribution of zeros and moments of the Riemann zeta-function, the prototypical example of an L-function, giving insight on previously unexplained, and in some cases previously unobserved, phenomena. The investigator also intends to combine tools from Fourier analysis with the theory of L-functions to study prime geodesics and to give improvements of several conditional estimates in prime number theory, assuming the generalized Riemann hypothesis.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.