Let λ be the Liouville function and P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence λ(P(n)) changes sign infinitely often. We present a solution to this problem for new classes of polynomials P, including any product of linear factors or any product of quadratic factors of a certain type. The proofs also establish some nontrivial cancellation in Chowla and Elliott type correlation averages.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
Related videos in this area: Analytic number theory, Chowla conjecture, Dirichlet L-functions, Dynamical systems, Ergodic theory, Number theory, Topology
