Higher uniformity of arithmetic functions in short intervals II. Almost all intervals

Authors

• Kaisa Matomäki, University of TurkuFollow
• Maksym Radziwiłł, Northwestern UniversityFollow
• Xuancheng Shao, University of KentuckyFollow
• Terence Tao, University of California, Los AngelesFollow
• Joni Teräväinen, University of CambridgeFollow

Abstract

We study higher uniformity properties of the von Mangoldt function Λ, the Möbius function μ, and the divisor functions dk on short intervals (x,x+H] for almost all x∈[X,2X]. Let Λ♯ and dk♯ be suitable approximants of Λ and dk, G/Γ a filtered nilmanifold, and F:G/Γ→C a Lipschitz function. Then our results imply for instance that when X1/3+ε≤H≤X we have, for almost all x∈[X,2X], (Formula presented.) for any fixed A>0, and that when Xε≤H≤X we have, for almost all x∈[X,2X], (Formula presented.) As a consequence, we show that the short interval Gowers norms ∥Λ−Λ♯∥Us(X,X+H] and ∥dk−dk♯∥Us(X,X+H] are also asymptotically small for any fixed s in the same ranges of H. This in turn allows us to establish the Hardy–Littlewood conjecture and the divisor correlation conjecture with a short average over one variable. Our main new ingredients are type II estimates obtained by developing a “contagion lemma” for nilsequences and then using this to “scale up” an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity.

Document Type

Article

Publication Date

6-1-2026

Notes/Citation Information

Publisher Copyright: © The Author(s) 2026.

Digital Object Identifier (DOI)

10.1007/s00222-026-01408-6

Archival?

Archival

Repository Citation

Matomäki, Kaisa; Radziwiłł, Maksym; Shao, Xuancheng; Tao, Terence; and Teräväinen, Joni, "Higher uniformity of arithmetic functions in short intervals II. Almost all intervals" (2026). Mathematics Faculty Publications. 54.
https://uknowledge.uky.edu/math_facpub/54