On the Liouville function at polynomial arguments
Joni Teräväinen (University of Oxford)
Wednesday 3rd March 16:00-17:00 online (zoom)
Let $\lambda$ 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 $\lambda(P(n))$ 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.