 Summatory totient function

The summatory totient function is $\Phi (x)={\sum }_{n\le x}\phi (n)$. It counts the coprime ordered pairs $(a,b)$ with $1\le a\le b\le x$. The identity ${\sum }_{d\mid n}\phi (d)=n$ yields ${\sum }_{j=1}^x\Phi (\lfloor x/j\rfloor )=x(x+1)/2$ for a positive integer $x$.