Jizhou Guo
Let $λ(n)=(-1)^{Ω(n)}$ be the Liouville function. We prove that there is an absolute constant $c>0$ such that, for every fixed $A>0$, $\displaystyle \sup_{1\le h\le(\log x)^A}\left|\sum_{n\le x}\frac{λ(n)λ(n+h)}{n}\right|\ll_A(\log x)^{1-c}.$ Thus a single power saving holds at every sufficiently large outer scale, uniformly throughout every fixed polylogarithmic shift range. The proof combines Pilatte's block decomposition with a flexible form of Menon's short-exponential-sum argument. The latter is proved here for every fixed power-logarithmic circle cutoff: we replay the minor arcs with all parameter dependencies exposed and localize the major arcs through residue classes and character-twisted short intervals. A one-sided typical-factorization truncation removes the sieve complement before Fourier inversion and hence introduces no loss in the shift. The remaining $h^{1/5}$ loss is cancelled by the flexible circle cutoff; a Hölder summation across the disjoint dyadic divisor blocks retains the resulting power saving. The centred term then follows from the arbitrary-interval decoupling theorem of Tao and Teräväinen. As a consequence, the same estimate holds uniformly for every pair of distinct shifts in a fixed polylogarithmic box. The exponent $A$ must be fixed before $x$ tends to infinity. The result is logarithmically weighted and does not prove the ordinary two-point Chowla conjecture.