On the Liouville function at polynomial arguments

TL;DR

This paper proves that the Liouville function applied to polynomial sequences changes sign infinitely often under broad conditions, extending previous conjectures and establishing new cancellation results for multiplicative functions.

Contribution

It demonstrates sign changes for $\lambda(P(n))$ for various classes of polynomials and proves a general cancellation result for bounded multiplicative functions composed with polynomials.

Findings

Sign changes of $\lambda(P(n))$ for polynomials with certain factorizations.

Nontrivial cancellation in partial sums of $g(P(n))$ for bounded multiplicative functions.

A multiplicative function analogue of a recent prime value result for polynomials.

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.