Evaluating von-Mangoldt sum of the form $\sum_{1\leq x\leq m} \sum_{1\leq y\leq n}\Lambda(x)\Lambda(y)$, for $m\neq n$

Question

I am unable to understand this since several months.

Let $Λ$ be the von-Mangoldt function.
Then we know the following: $\sum_{1\leq x\leq n}\Lambda(x)^{2}\sim n \log n$.

Then for $m\neq n$ and $x\neq y $ is the following true?\begin{equation}
\sum_{1\leq x\leq m} \sum_{1\leq y\leq n}\Lambda(x)\Lambda(y) \sim m n.
\end{equation}

Answer 1

Your sum is \begin{align*} & \left( {\sum\limits_{1 \le x \le m} {\Lambda (x)} } \right)\left( {\sum\limits_{1 \le y \le n} {\Lambda (y)} } \right) - \sum\limits_{1 \le z \le k} {\Lambda ^2 (z)} \\ & = (m + o(m))(n + o(n)) - k\log k+o(k\log k) \\ & = mn + o(mn) - k\log k+o(k\log k) =mn+o(mn), \end{align*} where $k=\min(m,n)$.