An averaged form of Chowla's conjecture

TL;DR

This paper proves an averaged version of Chowla's conjecture for the Liouville function, showing that the sum of correlations over shifts tends to zero on average, and provides exponential sum estimates with quantitative decay rates.

Contribution

The authors establish an averaged form of Chowla's conjecture using recent advances in multiplicative functions, extending results to more general functions and providing explicit decay bounds.

Findings

Proved averaged Chowla conjecture for the Liouville function.

Derived exponential sum estimates with explicit decay rates.

Extended results to more general bounded multiplicative functions.

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 $\alpha \in \mathbb{R}$, 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.