On the Liouville function at polynomial arguments

Abstract

abstract:

Let $\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne--Ferenczi--Mauduit--Rivat--S\'ark\"ozy asks to show that if $P(x)\in\mathbb{Z}[x]$, then the sequence $\lambda(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial.

We show that the sequence $\lambda(P(n))$ indeed changes sign infinitely often, provided that either (i) $P$ factorizes into linear factors over the rationals; or (ii) $P$ is a reducible cubic polynomial; or (iii) $P$ factorizes into a product of any number of quadratics of a certain type; or (iv) $P$ is any polynomial not belonging to an exceptional set of density zero.

Concerning (i), we prove more generally that the partial sums of $g(P(n))$ for $g$ a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on $g$. This establishes a ``99\% version'' of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of $g(P(n))$ and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value.