L-functions are the basic functions of number theory and are fundamental in the study of prime numbers, solutions to equations in whole numbers, and the uniformity of the distribution of arithmetic sequences. The first L-function is the Riemann zeta-function. The location of its zeros is the subject of the Riemann Hypothesis which is widely regarded as the most important unsolved problem in all of mathmatics. In this project the PI will study various approaches to the Riemann Hypothesis. In addition, he will investigate some very specific statistical properties of the Riemann zeta-function and of families of L-functions in order to more fully understand thes mysterious functions.
The PI and his collaborators intend to prove that most of the zeros of Dirichlet L-functions are on the critical line. They also will to develop a general tool, the asymptotic large sieve, to address problems involving averages over all primitive Dirichlet characters of modulus less than a given parameter. They will use this to investigate the spacings between zeros of Dirichlet L-functions. Another project is to determine (conjecturally) the arithemetic part of the distribution of spacings between consecutive zeros of the Riemann zeta-function. Finally, the PI would like to prove the reciprocity formula that he conjectured for Vasyunn sums, and to investigate its relevance in the Nyman-Beurling approach to the Riemann Hypothesis.
L-functions are some of the most important yet mysterious functions studied by mathematicians. This project was intended to penetrate some of the deeper properties of these L-functions from a statistical point of view. In particular there are recent conjectures arising from Random Matrix Theory (RMT) which predict in great detail how the values of these functions are distributed. In this project we proved a theorem about the sixth moment of a family known as Dirichlet L-functions. This work, which was joint with collaborators H. Iwaniec and K. Soundararajan, represents the strongest confirmation to date of the RMT predictions. To accomplish this result, we invented and developed something that we call the "asymptotic large sieve." This method has the potential for becoming a basic tool in analytic number theory. We have used it to study the zeros of Dirichlet L-functions and can show that at least 60% of these zeros are on the critical line (the vertical line that passes through the point 1/2). This result gives evidence toward the Generalized Riemann Hypothesis, which is a problem that many mathematicians view as one of the most important unsolved questions in all of mathematics. The PI for this grant was also involved in outreach projects, especially the Math Teachers' Circle program which brings together mathematicians and middle school math teachers together to build communities of problem solvers, and the Morgan Hill Math program which is a collection of after school enrichment programs for bright mathematics students in grades 4 - 12 who live in Morgan Hill, CA.
