The Chowla conjecture and Landau-Siegel zeroes

Abstract

Let $k\geqslant 2$ 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 $\sum_{n\leqslant x}\lambda(n+h_1)\cdots\lambda(n+h_k)=o(x)$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach. More precisely, we establish a bound for the sums $\sum_{n\leqslant x}\lambda(n+h_1)\cdots\lambda(n+h_k)$ under the existence of Landau-Siegel zeroes. 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.