Topics in Analytic Number Theory and Additive Combinatorics

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

This is an interdisciplinary project connecting number theory and combinatorics. One central theme in analytic number theory is to understand the distribution of primes numbers, a topic that has important applications in cryptography, coding theory, and financial security. It is a field that is currently experiencing intensive progress, in sometimes unexpected directions. In recent years, many important classical questions have seen spectacular advances based on new techniques; conversely, methods developed in analytic number theory have led to the solution of striking problems in other fields. The main goal of this research project is to make progress on the distributional properties of primes and other related arithmetic objects, by developing further the classical analytic tools and the more modern additive combinatorial tools, and by understanding how to effectively combine these two different tools together. It is hoped that the need for stronger results to cater for applications in analytic number theory will drive the development of additive combinatorics, and vice versa. The grant will also be used to help train graduate students in the area. More specifically, the PI will continue his work surrounding additive problems with primes. These works will involve developing both analytic tools and combinatorial tools. On the analytic side, the PI will continue to investigate the theory of Gowers norms for multiplicative functions and for the von-Mangoldt function, which are higher-order extensions to classical exponential sum estimates involving these functions. On the combinatorial side, the PI expects to develop tools from additive combinatorics for locating arithmetic structures within dense sets of integers and discover new mechanisms for using them to exhibit such structures in the primes. The additive combinatorial ideas behind these tools, such as pseudorandomness and Gowers uniformity norms, have a strong link with topics in theoretical computer science such as extractors and expanders, property testing, and error-correcting codes. The PI plans to explore these links in this project. This project is jointly funded by the Algebra and Number Theory program, the Established Program to Stimulate Competitive Research (EPSCoR), and the Combinatorics program. 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.