The Chowla conjecture and Landau-Siegel zeroes

TL;DR

This paper investigates the Chowla conjecture, providing conditional bounds on Liouville function sums assuming the existence of a Landau-Siegel zero, thus advancing understanding of correlations in multiplicative functions.

Contribution

It offers a new conditional bound on Chowla sums assuming Landau-Siegel zeroes, improving previous results by Germán and Kátai, Chinis, Tao, and Teräväinen.

Findings

Established non-trivial bounds under Landau-Siegel zero assumption

Improved upon previous results in the literature

Linked the behavior of Liouville sums to zeroes of L-functions

Abstract

Let $k\geq 2$ be an integer and let $\lambda$ be the Liouville function. Given $k$ non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n\leq x}\lambda(n+h_1)\cdots \lambda(n+h_k)=o(x)$ as $x\to\infty$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach towards it. More precisely, we establish a non-trivial bound for the sums $\sum_{n\leq x}\lambda(n+h_1)\cdots \lambda(n+h_k)$ under the existence of a Landau-Siegel zero for $x$ in an interval that depends on the modulus of the character whose Dirichlet series corresponds to the Landau-Siegel zero. Our work constitutes an improvement over the previous related results of Germ\'{a}n and K\'{a}tai, Chinis, and Tao and Ter\"av\"ainen.