Hardy-Littlewood conjecture predicts that the number of Goldbach decompositions $p+q=2n$ should be asymptotically equal to $K\frac{n}{\log^2 n}\prod\limits_{p>2,p\mid n}\frac{p-1}{p-2}$ for a positive $K$. This number of decompositions is, up to a negligible term, equal to the number of integers $r>0$ such that both $n-r$ and $n+r$ are prime. The latter is upper bounded by $f(n)=\sum\limits_{r=1}^{n-3}\frac{\Lambda(n-r)\Lambda(n+r)}{\log (n-r)\log (n+r)}$, itself lower bounded by $f_{low}(n)=\sum\limits_{r=1}^{n-3}\frac{\Lambda(n-r)\Lambda(n+r)}{\log^2 n}$ and upper bounded by $f_{up}(n)=\sum\limits_{r=1}^{n-3}\frac{\Lambda(n-r)\Lambda(n+r)}{\log 2n}$.

So does Hardy-littlewood conjecture imply $\sum\limits_{r=1}^{n-3}\Lambda(n-r)\Lambda(n+r)\sim 2C_{2}n$ where $C_{2}=.66016...$ is the twin prime constant?


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