## Abstract

Let A denote the Liouville function. We show that, as [Formula Presented] MATH for all fixed k and [Formula Presented] fixed but arbitrarily small. Previously this was only established for k < 1. We obtain this result as a special case of the corresponding statement for (non-pretentious) 1-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases e(-P (n)) by degree k nilsequences [Formula Presented]. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result [Formula Presented] in the same range of H. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of A over short polynomial progressions (n + P1 (m),…, n + Pk (m)), which in the case of linear polynomials yields a new averaged version of Chowla’s conjecture. We are in fact able to prove our results on polynomial phases in the wider range [Formula Presented] thus strengthening also previous work on the Fourier uniformity of the Liouville function.

Original language | American English |
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Pages (from-to) | 739-857 |

Number of pages | 119 |

Journal | Annals of Mathematics |

Volume | 197 |

Issue number | 2 |

DOIs | |

State | Published - 2023 |

### Bibliographical note

Publisher Copyright:© 2023 Department of Mathematics, Princeton University. All Rights Reserved.

## Keywords

- Chowla conjecture
- Gowers uniformity
- Liouville function
- nilsequences
- sign patterns