A1 Refereed original research article in a scientific journal
On the Liouville function at polynomial arguments
Authors: Teräväinen, Joni
Publisher: JOHNS HOPKINS UNIV PRESS
Publishing place: BALTIMORE
Publication year: 2024
Journal: American Journal of Mathematics
Journal name in source: AMERICAN JOURNAL OF MATHEMATICS
Journal acronym: AM J MATH
Volume: 146
Issue: 4
First page : 1115
Last page: 1167
Number of pages: 54
ISSN: 0002-9327
eISSN: 1080-6377
DOI: https://doi.org/10.1353/ajm.2024.a932436
Web address : https://muse.jhu.edu/article/932436
Self-archived copy’s web address: https://arxiv.org/pdf/2010.07924
Preprint address: https://arxiv.org/abs/2010.07924
Abstract
Let lambda denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-Sarkozy asks to show that if P(x) is an element of 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.
Let lambda denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-Sarkozy asks to show that if P(x) is an element of 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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