## Question about Combinatorial sieve

From the idea of Combinatorial sieve we know $$\sum_{\substack{d|(m,P_z) \ d \in D^{-}}} \mu(d) \leq \sum_{d|(m, P_z)} \mu(d)$$ and $$\sum_{\substack{d|(m+2,P_z) \ d \in D^{+}}} \mu(d) \geq \sum_{d|(m+2, P_z)} \mu(d)$$ my question is sum $$\sum_{m \in A} \left( \sum_{\substack{d|(m,P_z) \ d \in D^{-}}} \mu(d)\right) \left( \sum_{\substack{d|(m+2,P_z) \ d \in D^{+}}} \mu(d)\right)$$ can be a lower bound for numbers of elements $m \in A$ such that $m, m+2$ are both co-prime to $P_z$?

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 No, if $(m,P_z)=1$ and $(m+2,P_z)>1$, then the sum over $d\in D^-$ is equal to 1 and the sum over $d\in D^+$ is non-negative (and, in general, positive). – Dimitris Koukoulopoulos Nov 12 2011 at 7:40