Let $\Lambda$ be the von-Mangoldt function. What is the estimate for the sum is
$$\sum_{\substack{1\leq x\leq n\\1\leq y\leq n}}\Lambda(x)\Lambda(y)=\psi(n)\psi(n)\sim n^2.$$ $\psi$ is the Chebyshev function.
But in twin prime conjecture for a fixed integer $a$, $$\sum_{x\leq a} \Lambda(x)\Lambda(x+a)\sim C n$$ Here $C$ is the twin prime constant. I'm not able to distinguish the difference between two sums.
Can anyone help me out with this?