The Chowla conjecture and Landau-Siegel zeroes

TL;DR

This work establishes a substantial conditional bound for k-point correlations of the Liouville function under the existence of a Landau–Siegel zero. The authors develop a framework based on a z-parameterized model λ_z that captures the effect of a Siegel zero, prove a precise transition estimate between sums of λ and λ_z, and reformulate the main sum into quadratic-Dirichlet-character sums along a polynomial Q. They combine a beta-sieve approximation, Weil-type bounds for χ-sums, and Henriot’s multivariate divisor-sum bounds to control main and error terms, ultimately deriving a bound of the form ∑n≤x λ(n+h_1)…λ(n+h_k) ≪ xV/η + x exp(−c√(V log η)) for x=q^V with V in a specified range, thereby improving on prior results by Germán–Kátai, Chinis, and Tao–Teräväinen. The results illuminate how near-1 zeros of L-functions influence Chowla-type correlations and sharpen conditional progress toward Chowla’s conjecture in the Siegel-zero regime.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Corollary 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof