Zero-free regions for L-functions and related problems

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This award is for research in the theory of numbers. Every positive whole number is uniquely expressible as a product of primes. Primes are fascinating to study theoretically, but they also feature prominently in cryptography (the secure transmission of information). The distribution of primes is analytically encoded in the Riemann zeta function, the simplest example of an L-function. L-functions are ubiquitous in modern number theory. Many widely studied number-theoretic problems are naturally phrased in terms of properties of more general L-functions. This project will focus on non-vanishing of L-functions, individually and in parametric families. This is one of the most important questions regarding L-functions. For example, the distribution of zeros of the Riemann zeta function influences the distribution of primes (the subject of the Riemann Hypothesis), and conjecturally, the Hasse-Weil L-function of an elliptic curve vanishes at the point s = 1/2 if and only if the elliptic curve has infinitely many rational points (the Birch and Swinnerton-Dyer conjecture). The project includes training of undergraduate and graduate students. This project has three components. Towards the first component, the PI aims to develop new techniques to establish strong t-aspect zero-free regions for all Rankin-Selberg L-functions. The goal is a t-aspect zero-free region as strong as what de la Vallée Poussin established for the Riemann zeta function. Towards the second component, the PI aims to find new large classes of Rankin-Selberg L-functions for which one can establish a “hybrid-aspect” zero-free region with good t-dependence and no Landau-Siegel zero. This new uniformity will improve our understanding of the distribution of primes in relation to joint Sato-Tate laws involving two non-CM twist-inequivalent modular elliptic curves over a totally real number field. Towards the third component, the PI will continue earlier work on zero density estimates, showing that all L-functions in a family apart from a small exceptional set have very strong zero-free regions. 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.