On the Liouville function at polynomial arguments


Teräväinen, J

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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.

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Journal Article