Mathematics Colloquium

A major obstruction to proving the twin prime conjecture (but not bounded gaps between primes!) is the parity problem: the inability of sieve methods to detect the parity of the number of prime factors of integers in a sequence of interest. In the context of twin primes the parity barrier is captured by Chowla's conjecture on correlations of the Mobius function. Similar correlations are pervasive throughout analytic number theory, and there is at present no coherent set of tools to understand them. For instance, estimating correlations of multiplicative functions is often the key issue in the study of moments of L-functions or in subconvexity estimates. Until recently most such correlations (i.e those that are not related to GL(1) and GL(2) automorphic forms) were considered ``off-limits'' to methods of analytic number theory. But this is now changing. I will describe recent progress and the challenges that lie ahead. The talk will feature joint works with Chandee, Li, Matomaki and Tao.