Grant: Discovery Grants Program - Individual

While the study of prime numbers is an antique field of study, it is only recently that powerful tools have been developed to bring answers to century old questions. One of the main tools is the interplay between the primes and the zeros of the complex valued Riemann zeta function. The Riemann Hypothesis, a 150 year old conjecture, asserts that all its nontrivial zeros lie on the vertical 1/2-line. My research investigates the location and density of the zeros of the zeta function (and other L-functions) and the distribution of primes. It also has an explicit flavor which has become increasingly relevant. For instance, important work by Tao and Helfgott related to Goldbach's conjecture and other work by Hough solving Erdos covering congruences conjecture rely on explicit results about the primes. Moreover my results can be directly applied in other fields, such as Diophantine approximation, cryptography, and computer science. I first highlight a list of problems related to estimating some prime counting functions. Some of them concern improvements of works of respectively Rosser and Schoenfeld (for the primes) and of Ramare and Rumely (for the primes in arithmetic progressions). These are both still used frequently and referenced hundred of times. I recently proved new explicit bounds for psi(x) and psi(x;q,a) by developing an optimized smoothing method in conjunction with a sieving argument, and zero-free regions. I plan to establish explicit Burgess bounds for Dirichlet characters and to then deduce improved zero-free regions for Dirichlet L-functions. For psi(x), I also used the first explicit zero density estimate for zeta, and for psi(x;q,a), the extensive numerical verifications of Platt for the Generalized Riemann Hypothesis. Zero density estimates are a powerful tool as we are able to control how far and how few the zeros are from the 1-line. I plan to investigate further explicit bounds for them in the case of both the Riemann and Dirichlet L-functions. I also plan to bring some of these ideas in the study of primes in number fields. In the late seventies, Lagarias and Odlyzko gave effective versions of the Chebotarev Density Theorem and then, with Montgomery, provided upper bounds for the associated prime counting function Pi_C(x) for a larger range of x. These theorems relied on the distribution of the zeros of Dedekind zeta functions. There are now some explicit results concerning these (proven by myself or jointly with Ng). I am interested in extending these results to Hecke L-functions and to apply them to improve the bound Ng and I obtained for the least prime in the Chebotarev density. Moreover, I would like to prove explicit versions of some theorems of Serre, Wan, and Murty on Pi_C(x). Finally, I am interested in some applications including bounds for the prime counting functions of the Lang-Trotter conjecture.