This project concerns research in the Theory of Numbers, in particular questions related to the distribution of prime numbers. This topic has advanced considerably during the past year due to the breakthrough work of Zhang and Maynard on small gaps between consecutive prime numbers, where previous work of Fouvry, Iwaniec, and Friedlander played a crucial role. One of the most fundamental open questions in modern mathematics is the Riemann hypothesis, which connects the distribution of prime numbers with the distribution of complex zeros of the generating zeta function. The hypothesis asserts that all these zeros lie on a single "critical" line. The main goal of this project is to show that a large proportion of these zeros are indeed resting there. This result will enhance our understanding of subtle behavior of the basic elements of arithmetic, the prime numbers. The PI has written expository books on several topics related to the Theory of Numbers, most recently a book on Sieve Methods, joint with John Friedlander. The PI is active in training graduate students and plans to continue to train graduate students under this award.
The project starts with the method of Levinson-Conrey designed for estimating the percentage of zeros of the Riemann zeta function that are on the critical line. The original construction by Conrey of a linear combination of the zeta function and its odd order derivatives is quite robust. Naturally, this needs to be mollified by a suitable Dirichlet polynomial. The longer mollifier one can handle, the more critical zeros reveal. The key innovation of the proposed approach is a special arrangement of the convolution coefficients into "local mollifiers" which allows applying very long global mollification, thus raising hope for substantial improvements of the existing results.
