A1 Refereed original research article in a scientific journal

An averaged form of Chowla's conjecture




AuthorsKaisa Matomäki, Maksym Radziwill, Terence Tao

PublisherMathematical Sciences Publisher

Publishing placeBerkeley

Publication year2015

JournalAlgebra and Number Theory

Volume9

Issue9

First page 2167

Last page2196

Number of pages30

ISSN1937-0652

eISSN1944-7833

DOIhttps://doi.org/10.2140/ant.2015.9.2167


Abstract

Let $\lambda$ denote the Liouville function.  A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,\dots,h_k$, one has $\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k) = o(X)$ as $X \to \infty$.  This conjecture remains unproven for any $h_1,\dots,h_k$ with $k \geq 2$.  In this paper, using the recent results of the first two authors on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain-Sarnak-Ziegler, we establish an averaged version of this conjecture, namely

$$ \sum_{h_1,\dots,h_k \leq H} \left|\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k)\right| = o(H^kX)$$

as $X \to \infty$ whenever $H = H(X) \leq X$ goes to infinity as $X \to \infty$, and $k$ is fixed.  Related to this, we give the exponential sum estimate

$$ \int_0^X \left|\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)\right| dx = o( HX )$$

as $X \to \infty$ uniformly for all $\alpha \in \R$, with $H$ as before.  Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of $\frac{\log\log H}{\log H}$), and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

 



Downloadable publication

This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail. Please cite the original version.





Last updated on 2024-26-11 at 22:42