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 is unweighted, and the estimate must hold at every scale. Make the following constructions and their parameter choices precise within the proof: Singleton labels are prime factors or labels occurring exactly once in composite steps or graph walks. Establish exact cancellation of the signed terms they supply. Use a low-rank forest encoding to describe repeated prime occurrences in composite walks. Establish nonnegative support of the low-rank part sufficient to control matrix trace expansions. Let (P) denote the system of prime families or prime sets under consideration. Use truncation bounds and cutoff weights to control parameter losses from sparse prime divisors and large divisor counts. Respect the finite Chinese remainder theorem obstruction to uniform mixing over (P) in cutoff estimates. Witnesses are divisibility and deletion predicates that cover singleton primes through triangular congruence elimination. Witness-exclusion Boolean variables encode whether each witness condition holds or is excluded. Establish the full mixed-difference property, stronger than pairwise changes. Expand the combinatorial product over all potential witness-exclusion Boolean variables. Prove that a nonzero total mixed difference forces at least one intersection of witnesses within the same hybrid to cover all singleton labels. When no such intersection exists, prove exact term-by-term cancellation. Hybrids are finite, deterministic transition states in a hybrid argument chain. Use Braverman’s circuit independence theorem to transfer independent-residue calculations to finite intervals, verifying its hypotheses in this setting. Select the required covering hybrid by a finite deterministic procedure without exhaustive combinatorial counting.