Interplay of multiplicative number theory and additive combinatorics

This project concerns multiplicative number theory and its interplay with the emerging topic of additive combinatorics. Multiplicative number theory is an area of number theory concerned with the study of prime numbers and multiplicative functions. One of the most important unsolved questions in this area and in all of number theory is the twin prime conjecture, asserting that there are infinitely many pairs of prime numbers differing by two. In 1965, Chowla formulated an influential conjecture that can be viewed as an approximation to the twin prime conjecture. Chowla’s conjecture predicts that the prime factorisations of consecutive integers behave independently of each other. This conjecture captures the key difficulty in the twin prime conjecture, but yet there has been a lot of recent progress on Chowla’s conjecture by the applicant and others. The aim of this project is to make substantial progress on Chowla’s conjecture, as well as on other key questions in multiplicative number theory, using a mixture of methods from analytic number theory and additive combinatorics, as well as higher order Fourier analysis, a theory recently developed by Green and Tao. Connections between Chowla’s conjecture and questions in additive combinatorics and higher order Fourier analysis have recently been discovered in works of the applicant and others, and the proposed research aims at exploiting these connections to make substantial progress on Chowla’s conjecture. The project also involves several other problems of interest in number theory, such as the Hardy—Littlewood conjecture on average and the Hasse principle for almost all surfaces of a certain type.