Logarithmic Chowla Correlations Across All Shift Scales

Abstract

Let $\lambda (n)=(-1{)}^{\Omega (n)}$ be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute $c>0$ such that every sufficiently large $x$ admits a single set ${\mathcal{E}}_x\subseteq [1,x]$ with $|{\mathcal{E}}_x\cap [1,H]|{\ll }_{A}H(\log x{)}^{-A}$ $(1\le H\le x)$ for every fixed $A>0$, while ${\max }_{\begin{array}{c}1\le h\le xh\notin {\mathcal{E}}_x\end{array}}{\sup }_{1\le y\le x}\left|{\sum }_{n\le y}\frac{\lambda (n)\lambda (n+h)}{n}\right|\ll (\log x{)}^{1-c}$. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of $\log x$. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\in \mathbb{N}$ and $1\le y\le x$, $\left|{\sum }_{n\le y}\frac{\lambda (n)\lambda (n+h)}{n}\right|\le \log (2\min h,y)+O((\log x{)}^{1-c_{\mathrm{G}}})$ for an absolute $c_{\mathrm{G}}>0$, with no exceptional shifts.

arXiv comment84 pages. Major revision with strengthened main results and expanded proofs