Summatory totient function
= Summatory totient function
{title2=$\Phi(x)$}
= Totient summatory function
{synonym}
= Summatory totient
{synonym}
The summatory totient function is $\Phi(x)=\sum_{n\le x}\varphi(n)$. It counts the <coprime> <ordered pairs> $(a,b)$ with $1\le a\le b\le x$. The identity $\sum_{d\mid n}\varphi(d)=n$ yields $\sum_{j=1}^{x}\Phi(\lfloor x/j\rfloor)=x(x+1)/2$ for a positive <integer> $x$.