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.
Categories: math.NT math.CO