Fourier uniformity of bounded multiplicative functions in short intervals on average

Kaisa Matomäki, Maksym Radziwiłł*, Terence Tao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

Let λ denote the Liouville function. We show that as X→ ∞, ∫X2Xsupα|∑x<n≤x+Hλ(n)e(-αn)|dx=o(XH)for all H≥ Xθ with θ> 0 fixed but arbitrarily small. Previously, this was only known for θ> 5 / 8. For smaller values of θ this is the first “non-trivial” case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) 1-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of λ(n) Λ (n+ h) Λ (n+ 2 h) over the ranges h< Xθ and n< X, and where Λ is the von Mangoldt function.

Original languageEnglish (US)
Pages (from-to)1-58
Number of pages58
JournalInventiones Mathematicae
Volume220
Issue number1
DOIs
StatePublished - Apr 1 2020

ASJC Scopus subject areas

  • General Mathematics

Fingerprint

Dive into the research topics of 'Fourier uniformity of bounded multiplicative functions in short intervals on average'. Together they form a unique fingerprint.

Cite this