Sieves and primes

Questions about properties of the positive integers have fascinated people for thousands of years and have recently found applications in computer science, information security and signal processing. In this field, known as number theory, the prime numbers play a central role. Fundamental questions revolve around patterns in the primes, gaps between primes, and how primes of special types are distributed. Since the early 20th century, sieve methods have been one of the chief tools we have for analyzing these problems, but the limitations of these methods are poorly understood, and discovering the limitations is a major open problem in the field. This award will enable the PI to continue his work understanding and exploring sieve methods and the distribution of primes. Success in this endeavor will help unlock many of the mysteries of prime numbers and have a significant impact on many areas of mathematics and information theory. Grant funds will also be used to train and mentor graduate students working in number theory. The PI will develop new methods of probing the limitations of sieve methods for detecting primes in a general sequence of integers. The emphasis will be on developing a new, unified theory of sieves that allows one to say if the main hypotheses on the sequence, known as Type-I bounds and Type-II bounds, are sufficient to show that the sequence contains many primes. The strength of these hypotheses are governed by three parameters. In particular, we will prove, for the first time, that in a certain range of these parameters, there are sequences which satisfy the Type-I and Type-II bounds yet contain no primes. The primary goal is to determine precisely in which range of the parameters the main hypotheses imply that the sequence always contains many primes. We will also investigate in finer detail the problem of counting primes in short intervals. The PI will also continue his investigations into further understanding the concentration of divisors of integers. The primary goal is to determine precisely the measure of the concentration function of divisors of typical integers. 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.