On the Liouville function at polynomial arguments

abstract: Let 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), then the sequence (P (n) ) changes sign infinitely often, assuming only that P (x) is not the square of another polynomial. We show that the sequence (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.