On the Liouville function at polynomial arguments
Let λλ be the Liouville function and P(x)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))λ(P(n)) changes sign infinitely often. We present a solution to this problem for new classes of polynomials PP, 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.