Improved bounds for the two-point logarithmic Chowla conjecture

Source abstract

Let λ \lambda be the Liouville function, defined as λ ( n ) ≔ ( − 1 ) Ω ( n ) \lambda (n) ≔(-1)^{\Omega (n)} where Ω ( n ) \Omega (n) is the number of prime factors of n n with multiplicity. In 2021, Helfgott and Radziwiłł proved that ∑ n ⩽ x 1 n λ ( n ) λ ( n + 1 ) ≪ log ⁡ x ( log ⁡ log ⁡ x ) 1 / 2 , $$\begin{equation*} \sum _{n\leqslant x} \frac {1}{n}\lambda (n) \lambda (n+1) \ll \frac {\log x}{(\log \log x)^{1/2}}, \end{equation*}$$ improving earlier results by Tao and Teräväinen. We prove that ∑ n ⩽ x 1 n λ ( n ) λ ( n + 1 ) ≪ ( log ⁡ x ) 1 − c $$\begin{equation*} \sum _{n\leqslant x} \frac 1n \lambda (n) \lambda (n+1) \ll (\log x)^{1-c} \end{equation*}$$ for some absolute constant c > 0 c>0 . This appears to be best possible with current methods.