Improved bounds for the two-point logarithmic Chowla conjecture

TL;DR

This paper improves the upper bounds for the two-point logarithmic Chowla conjecture, showing the sum involving the Liouville function and its shift is bounded by a power of log x, which is likely optimal with current techniques.

Contribution

The authors establish a sharper bound for the sum of the Liouville function multiplied by its shift, advancing the understanding of correlations in multiplicative functions.

Findings

Bounded the sum by a power of log x with a positive constant c

Achieved a bound that is likely optimal with current methods

Extended previous results by Helfgott, Radziwi{
42}{
42}, Tao, and Ter"av"ainen

Abstract

Let $\lambda$ be the Liouville function, defined as $\lambda(n) := (-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime factors of $n$ with multiplicity. In 2021, Helfgott and Radziwi{\l}{\l} proved that $$\sum_{n\leq x} \frac{1}{n} \lambda(n) \lambda(n+1) \ll \frac{\log x}{(\log \log x)^{1/2}},$$improving earlier results by Tao and Ter\"av\"ainen. We prove that $$\sum_{n\leq x} \frac{1}{n} \lambda(n) \lambda(n+1) \ll (\log x)^{1-c}$$for some absolute constant $c>0$. This appears to be best possible with current methods.