Analytic number theory and higher order Fourier analysis

Many of the central problems in analytic number theory concern the existence of patterns in the primes or the randomness properties of the Mbius function.

The aim of this proposal is to make progress on such questions, using in particular methods from higher order Fourier analysis.Two fundamental conjectures about randomness of the Mbius function are the Chowla and Sarnak conjectures.

This proposal aims to make progress on these conjectures and their variants, in particular by studying a conjecture on higher order uniformity of the Mbius function that is closely tied to these problems.Regarding patterns in the primes, the approach in this proposal is to establish new Gowers norm estimates for the primes and other multiplicatively defined sequences.

Both quantitative Gowers norm bounds and short interval Gowers norms will be studied.

These have potential for various applications, such as a version of the prime tuples conjecture with a short average and convergence results for multiple ergodic averages along the primes.