Fast algorithms, computational complexity, and subconvexity bounds in analytic number theory

Ohio State University, The, Columbus OH

Investigators

Abstract

The topics in this proposal fall within the area of number theory. This is a fundamental area in mathematics interested in understanding the integers. There is deep information about the integers encoded in certain analytic objects called L-functions. And there have been many fruitful interactions between L-functions and computational methods. Such interactions started with the Riemann zeta function which, famously, Riemann computed numerically. The role of computation in analytic number theory has continued to grow since then. The proposer will investigate certain L-functions from a computational viewpoint. This includes investigating their computational complexity, the derivation of new fast algorithms, with an application to learning about integer factorization, and the use of computation as an experimental tool in the study of L-functions. The proposer will study several independent topics in analytic and computational number theory. The main topic is to investigate connections between subconvexity estimates of the Riemann zeta function, on the one hand, and its computational complexity on the other. Part of the novelty here is that these two aspects are weakly related in general. However, they connect strongly for the Riemann zeta function in the critical strip, and also for Dirichlet character sums to a power-full modulus, perhaps even for more number-theoretic objects. One goal of the project is to translate recent progress between the analytic and computational viewpoints. Another topic concerns the full moment conjectures of ``families'' of L-functions with symplectic or orthogonal symmetry. The proposer aims to derive uniform asymptotics for the full moment conjectures for such families, extending previous joint work with Michael Rubinstein in the unitary case. A third topic is to develop the recent algorithm for detecting squarefree numbers that was derived in joint work with Andrew Booker and Jon Keating, especially in relation to lower bounds for the growth rate of character sums over the primes.