The Chowla conjecture and Landau–Siegel zeroes

Abstract

<p>Let k⩾2k⩾2 be an integer and let λλ be the Liouville function. Given <em>k</em> non-negative distinct integers h1,…,hkh1,…,hk, the Chowla conjecture claims that ∑n⩽xλ(n+h1)⋯λ(n+hk)=o(x)∑n⩽xλ(n+h1)⋯λ(n+hk)=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 non-trivial bound for the sums ∑n⩽xλ(n+h1)⋯λ(n+hk)∑n⩽xλ(n+h1)⋯λ(n+hk) under the existence of a Landau–Siegel zero for <em>x</em> 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.</p>