I am not able to prove that $$\left(3+4\sum_{k=2}^N\phi(k)\right)^2 \sim \frac{12^2}{\pi^4}N^4 \quad (N\to+\infty).$$ Any idea?
Curiosity: LSH is the cardinality of the subset of the $\mathbb{Q}^2$ of couples $(m/n,p/q)$ such that $\lvert m \rvert, \lvert n \rvert, \lvert p \rvert, \lvert q \rvert \leq N$.