Sign Patterns of the Mobius Function

This lecture explores the Mobius function, one of the most significant arithmetic functions in number theory, focusing on its randomness properties and sign patterns. Delve into the "Mobius randomness law" principle which suggests the Mobius function should be orthogonal to any "structured" sequence. Learn about P. Sarnak's influential conjecture that formalizes this principle by proposing that "structured sequences" correspond to those arising from deterministic dynamical systems. Discover how Sarnak's conjecture relates to Chowla's conjecture (the Mobius version of the prime tuple conjecture) and examine recent progress toward proving these conjectures through advances in dynamics, additive combinatorics, and analytic number theory.