Let (\lambda(n)=(-1)^{\Omega(n)}), where (\Omega(n)) counts ...

Prompt

Let (\lambda(n)=(-1)^{\Omega(n)}), where (\Omega(n)) counts prime factors with multiplicity. Prove that an absolute constant (c>0) exists such that, for every fixed pair of affine forms (L_j(n)=a_jn+b_j), (j=1,2), with (a_j\in\mathbb Z_{>0}), (b_j\in\mathbb Z_{\ge0}), and (a_1b_2-a_2b_1\ne0), [ \left|\sum_{1\le n\le X}\lambda(L_1(n))\lambda(L_2(n))\right| \ll_{a_1,a_2,b_1,b_2}\frac{X}{(\log X)^c} \qquad\text{for every }X\ge3. ] The sum must be unweighted, and the estimate must hold at every scale. Proof requirements: Establish exact singleton cancellation and nonnegative support of the low-rank part. Respect the finite Chinese remainder theorem restriction against mixing over (P) when handling cutoffs. Establish the full mixed-difference property, stronger than pairwise changes: if the total mixed difference is nonzero, expand the product over all potential witness-exclusion Boolean variables. Show that at least one intersection of witnesses within the same hybrid covers all singleton labels. Prove term-by-term cancellation whenever no such intersection exists. Choose the hybrid by a finite deterministic procedure, avoiding exhaustive combinatorial counting