Log-free zero-density estimates for the Riemann zeta-function

Description:

Much of our knowledge of low-lying Riemann zeta zeroes comes from finite verifications of the Riemann hypothesis and the application of zero-free regions. Unfortunately, these are not sufficient for many applications to the distribution of primes. Very few classical zero-density estimates have been made explicit. Those that have only start to show their teeth at very high heights. This is due to the presence of some log-factors, which, while insignificant for large height, are non-negligible at the heights of interest for applications. This project will investigate making explicit some log-free zero-density estimates, and applying these to problems involving prime number distribution.

This project would align with the current interests of the number theory group at the School of Science, UNSW Canberra.