Class Groups of Number Fields and Zeros of L-functions

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Number theory is the branch of mathematics concerned with studying the integers, and more specifically, the primes. The Prime Number Theorem (which describes the distribution of the primes among the positive integers) was proved in 1896 independently by Hadamard and de la Vallee Poussin by understanding certain properties of the Riemann zeta-function. The Riemann zeta-function and its generalizations, called L-functions, are ubiquitous yet mysterious functions in number theory. These functions can be defined in association with a plethora of mathematical objects, including Dirichlet characters, number fields, and elliptic curves. Understanding the location of the zeros of L-functions is a central problem in all of mathematics. While we cannot presently prove the Riemann Hypothesis, posed by Riemann in 1859, there are many fruitful investigations to pursue to better understand the zeros of L-functions. In particular, the vertical distribution of the zeros of L-functions has deep connections to two other central problems: the class number problem, which has its beginnings in the work of Gauss, and the possibility of a special type of counterexample to the (Generalized) Riemann Hypothesis. These hypothetical counterexamples are called Landau-Siegel zeros, and presently their existence cannot be ruled out.More specifically, this project will pursue problems in the intersection of analytic and algebraic number theory. It will study applications related to the Chebotarev density theorem for families of L-functions, the vertical distribution of zeros of L-functions, and class numbers of number fields. The activities of the project will also have broader impacts in terms of mentoring both graduate and undergraduate students in a liberal arts setting.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.