A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
On the Liouville function at polynomial arguments
Tekijät: Teräväinen, Joni
Kustantaja: JOHNS HOPKINS UNIV PRESS
Kustannuspaikka: BALTIMORE
Julkaisuvuosi: 2024
Journal: American Journal of Mathematics
Tietokannassa oleva lehden nimi: AMERICAN JOURNAL OF MATHEMATICS
Lehden akronyymi: AM J MATH
Vuosikerta: 146
Numero: 4
Aloitussivu: 1115
Lopetussivu: 1167
Sivujen määrä: 54
ISSN: 0002-9327
eISSN: 1080-6377
DOI: https://doi.org/10.1353/ajm.2024.a932436
Verkko-osoite: https://muse.jhu.edu/article/932436
Rinnakkaistallenteen osoite: https://arxiv.org/pdf/2010.07924
Preprintin osoite: https://arxiv.org/abs/2010.07924
Tiivistelmä
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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