Improved bounds for the two-point logarithmic Chowla conjecture

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 Radziwiłł proved that $$\sum _{n\le x}\frac{1}{n}\lambda (n)\lambda (n+1)\ll \frac{\log x}{(\log \log x{)}^{1/2}},$$improving earlier results by Tao and Teräväinen. We prove that $$\sum _{n\le x}\frac{1}{n}\lambda (n)\lambda (n+1)\ll (\log x{)}^{1-c}$$for some absolute constant $c>0$. This appears to be best possible with current methods.

Minor typos corrected. 66 pages