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2015 An averaged form of Chowla's conjecture
Kaisa Matomäki, Maksym Radziwiłł, Terence Tao
Algebra Number Theory 9(9): 2167-2196 (2015). DOI: 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

$\sum _{1\le n\le X}\phantom{\rule{0.3em}{0ex}}\lambda \left(n+{h}_{1}\right)\cdots \lambda \left(n+{h}_{k}\right)=o\left(X\right)$

as $X\to \infty$. This conjecture remains unproven for any ${h}_{1},\dots ,{h}_{k}$ with $k\ge 2$. Using the recent results of Matomäki and Radziwiłł on mean values of multiplicative functions in short intervals, combined with an argument of Kátai and Bourgain, Sarnak, and Ziegler, we establish an averaged version of this conjecture, namely

$\sum _{{h}_{1},\dots ,{h}_{k}\le H}|\sum _{1\le n\le X}\phantom{\rule{0.3em}{0ex}}\lambda \left(n+{h}_{1}\right)\cdots \lambda \left(n+{h}_{k}\right)|=o\left({H}^{k}X\right)$

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

${\int }_{0}^{X}|\sum _{x\le n\le x+H}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\lambda \left(n\right)e\left(\alpha n\right)|\phantom{\rule{0.3em}{0ex}}dx=o\left(H“X\right)$

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 $loglogH∕logH$) and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

## Citation

Kaisa Matomäki. Maksym Radziwiłł. Terence Tao. "An averaged form of Chowla's conjecture." Algebra Number Theory 9 (9) 2167 - 2196, 2015. https://doi.org/10.2140/ant.2015.9.2167

## Information

Received: 17 April 2015; Revised: 28 August 2015; Accepted: 6 October 2015; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1377.11109
MathSciNet: MR3435814
Digital Object Identifier: 10.2140/ant.2015.9.2167

Subjects:
Primary: 11P32

Keywords: Chowla conjecture , Hardy–Littlewood circle method , multiplicative functions