An averaged form of Chowla’s conjecture

Let $λ$ 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} λ (n + h_{1}) \dots λ (n + h_{k}) = o (X)

as $X \to \infty$ . This conjecture remains unproven for any $h_{1}, \dots, h_{k}$ with $k \geq 2$ . 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} \leq H} | \sum_{1 \leq n \leq X} λ (n + h_{1}) \dots λ (n + h_{k}) | = o (H^{k} X)

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} | \sum_{x \leq n \leq x + H} λ (n) e (α n) | d x = o (H “ X)

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

Keywords

multiplicative functions, Hardy–Littlewood circle method, Chowla conjecture

Mathematical Subject Classification 2010

Primary: 11P32

Milestones

Received: 17 April 2015

Revised: 28 August 2015

Accepted: 6 October 2015

Published: 4 November 2015

Authors

Kaisa Matomäki
Department of Mathematics and Statistics University of Turku FI-20014 Turku Finland

Maksym Radziwiłł
Department of Mathematics Rutgers University Hill Center for the Mathematical Sciences 110 Frelinghuysen Rd. Piscataway, NJ 08854-8019 United States

Terence Tao
Department of Mathematics University of California Los Angeles 405 Hilgard Avenue Los Angeles, CA 90095-1555 United States

Vol. 9, No. 9, 2015

Kaisa Matomäki, Maksym Radziwiłł and Terence Tao

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors