Breaking barriers with large sieve inequalities

This award concerns research in number theory. The study of arithmetic sequences over prime numbers and the value of L-functions have been foundational topics of research in number theory over the past two centuries. Large sieve inequalities, which estimate the number of integers which remain after removing a set of residue classes modulo certain primes, are an important tool used to tackle such problems. Often, the strength of the relevant large sieve inequality dictates the progress one can make. For instance, Kummer in the 19th century (refined later Patterson in the 20th century) predicted that an important exponential sum that arises naturally in arithmetic geometry exhibits a subtle statistical bias over the primes. This problem was studied on some of the first super computers in the 1950s and 60s. The PI and his collaborators recently explained this bias with surprising new insights on the relevant large sieve ensemble. In this project, the PI will seek to push the boundaries on large sieve inequalities for families of harmonics that are perceived right now to be "stuck", or right on the edge of current technology. The PI and collaborators will explore the potential consequences of their methods for moments and zeros of L-functions, non-vanishing of central values of L-functions, and bounds for exponential sums over primes. Broader impact of this project includes the training of students and postdocs and organizing seminars. An influential conjecture of Chowla asserts that all primitive Dirichlet L-functions do not vanish at the central point. The PI plans to leverage the under utilised connection between metaplectic forms and fixed order characters over number fields to make progress toward Chowla's conjecture. The PI and his collaborators discovered that such a connection was a key ingredient in the resolution of Patterson's conjecture because it explained why a certain large sieve inequality was not optimal. The PI will extend these ideas to higher order characters, where much less is known, and will use techniques from analytic number theory to study the non-vanishing of central values of L-functions and moments of Hecke L-functions. The PI and collaborators will also use trace formulae to study large sieve inequalities for families that arise in moments of automorphic L-functions. 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.