# Does the first Hardy-Littlewood conjecture imply $\sum\limits_{r=1}^{n-3}\Lambda(n-r)\Lambda(n+r)\sim 2C_{2}n$?

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?