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Let λ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-Sárközy asks to show that if P(x) ∈ Z[x], then the sequence λ(P(n)) changes sign infinitely often, assuming only that P(x) is not the square of another polynomial. We show that the sequence λ(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. [ABSTRACT FROM AUTHOR]

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs. As a first application, we show that for a multiplicative function ƒ : N → {−1,1}, lim supx→∞ |∑n≤xμ2(n)ƒ(n)| = ∞. This confirms a conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions. Secondly, we show that a completely multiplicative function ƒ : N → C satisfies ∑n≤xƒ(n) = cx + O(1)d with c ≠ 0 if and only if ƒ(p) = 1 for all but finitely many primes and |ƒ(p)| < 1 for the remaining primes. This answers a question of Ruzsa. For the case c = 0, we show, under the additional hypothesis ∑p 1−|ƒ(p)|/p < ∞, that ƒ has bounded partial sums if and only if ƒ(p) = χ (p)pit for some non-principal Dirichlet character χ modulo q and t ∈ R except on a finite set of primes that contains the primes dividing q, wherein |f(p)| < 1. This provides progress on another problem of Ruzsa and gives a new and simpler proof of a stronger form of Chudakov's conjecture. Along the way we obtain quantitative bounds for the discrepancy of the modified characters improving on the previous work of Borwein, Choi and Coons. [ABSTRACT FROM AUTHOR]