The main goal of this project is to refine the recent method of Goldston, Pintz, and Yildirim in several specific problems. The long-term goal is to investigate some of the more difficult questions for which no progress has yet been made. The method of Goldston, Pintz and Yildirim detects primes or almost primes close together by examining these sequences when weighted by certain sieve functions which are large when there are primes in specified configurations. These sieve functions can be obtained by the Selberg sieve or by an optimization process. Some questions to be examined are to find the smallest prime gap size that can be obtained unconditionally by the method, to determine how the sieve weights behave depending on the number of prime factors a number has, and to examine certain sequences of primes for positive proportion results. The investigator will also continue his work on jumping champions with A. Ledoan, work on error terms in additive prime number theory, and examine applications of higher correlation of zeros of zeta functions to obtaining information on multiplicity and gaps between zeros.
This project is concerned with prime numbers, which have been studied by mathematicians for thousands of years, and in recent years have found important applications in computer science and cryptography. Despite the simplicity of how they arise, prime numbers offer some of the most challenging and difficult problems in mathematics, including the Riemann Hypothesis (which is equivalent to the primes having a very regular distribution) and the Goldbach and Twin Prime Conjectures. The study of these questions has contributed to the development of not just number theory but analysis and algebra. More generally, number theory has applications throughout mathematics and fields that make use of mathematics.
