This project involves research in the areas of analytic number theory and additive combinatorics. An important class of problems in these areas concern the distribution of prime numbers, a topic that has had important applications in theoretical computer science, cryptography, and financial security. One of the main goals 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.
More specifically, the PI will continue his work surrounding additive problems involving primes by developing and combining aforementioned analytic and combinatorial tools. The classical approaches using the theory of L-functions and sieve theory have achieved great results in this direction, and it is expected that by incorporating tools from additive combinatorics one can go beyond what the classical methods could achieve. The PI will also continue to investigate the theory of Gowers norms for multiplicative functions and for the von-Mangoldt function. These are "higher-order" extensions to classical exponential sum estimates involving these functions. The study of general Gowers norms for these functions is an essential analytic input for applying additive combinatorial tools.
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.
