The Chowla conjecture and Landau-Siegel zeroes

Abstract

Let $k\geq 2$ be an integer and let $λ$ be the Liouville function. Given $k$ non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n\leq x}λ(n+h_1)\cdots λ(n+h_k)=o(x)$ 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 $\sum_{n\leq x}λ(n+h_1)\cdots λ(n+h_k)$ 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 Germán and Kátai, Chinis, and Tao and Teräväinen.