The Chowla conjecture and Landau-Siegel zeroes

Abstract

Let $k\ge 2$ be an integer and let $\lambda$ be the Liouville function. Given $k$ non-negative distinct integers $h_1,\dots ,h_k$, the Chowla conjecture claims that ${\sum }_{n\le x}\lambda (n+h_1)\cdots \lambda (n+h_k)=o(x)$ 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 ${\sum }_{n\le x}\lambda (n+h_1)\cdots \lambda (n+h_k)$ 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 Germán and Kátai, Chinis, and Tao and Teräväinen.

20 pages; Published online in Math. Proc. Camb. Phil. Soc.; The proof of Theorem 1.1 (Section 5) was divided into subsections for easier reading, minor text changes and corrections were applied, and a reference was added

References in corpus (2)

• Siegel Zeros and Sarnak's Conjecture
• The Hardy--Littlewood--Chowla conjecture in the presence of a Siegel zero

Cited by in corpus (1)

• On variants of Chowla's conjecture