On the Liouville function at polynomial arguments

On the Liouville function at polynomial arguments

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Joni Teravainen, University of Oxford

Zoom link:  https://princeton.zoom.us/j/97126136441

Password: the three digit integer that is the cube of the sum of its digits


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.