On the Liouville function at polynomial arguments

Abstract

Let $\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-Sárközy 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.

43 pages; further referee comments incorporated; to appear in Amer. J. Math

References in corpus (6)

• Higher Descent on Pell Conics I. From Legendre to Selmer
• Multiplicative functions that are close to their mean
• The parity problem for reducible cubic forms
• Schinzel Hypothesis on average and rational points
• The parity problem for irreducible cubic forms
• Infinitely many sign changes of the Liouville function on $x^2+d$

Cited by in corpus (2)

• Partition regularity of Pythagorean pairs
• On variants of Chowla's conjecture