Vol. 13, No. 9, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 8, 1777–2003
Issue 7, 1547–1776
Issue 6, 1327–1546
Issue 5, 1025–1326
Issue 4, 777–1024
Issue 3, 521–775
Issue 2, 231–519
Issue 1, 1–230

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

Terence Tao and Joni Teräväinen

Vol. 13 (2019), No. 9, 2103–2150
Abstract

We study the asymptotic behaviour of higher order correlations

EnXdg1(n + ah1)gk(n + ahk)

as a function of the parameters a and d, where g1,,gk are bounded multiplicative functions, h1,,hk are integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all X if g1gk does not (weakly) pretend to be a twisted Dirichlet character nχ(n)nit, and behave asymptotically like a multiple of d itχ(a) otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the d parameter is averaged out and one can set t = 0. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the k-point Chowla conjecture EnXλ(n + h1)λ(n + hk) = o(1) for k odd or equal to 2 for all scales X outside of a set of zero logarithmic density.

Keywords
multiplicative functions, Chowla's conjecture, Elliott's conjecture
Mathematical Subject Classification 2010
Primary: 11N37
Secondary: 37A45
Milestones
Received: 8 September 2018
Revised: 24 June 2019
Accepted: 24 July 2019
Published: 7 December 2019
Authors
Terence Tao
Department of Mathematics
University of California Los Angeles
Los Angeles, CA
United States
Joni Teräväinen
Mathematical Institute
University of Oxford
United Kingdom