Limit to Infinity of Summation of Euler Phi Function over Square

Theorem

 * $\ds \lim_{n \mathop \to \infty} \dfrac {\map \Phi n} {n^2} = \dfrac 3 {\pi^2}$

where:


 * $\map \Phi n = \ds \sum_{k \mathop = 1}^n \map \phi k$
 * $\map \phi k$ is the Euler $\phi$ function of $k$.

Numerically, this evaluates to:


 * $\dfrac 3 {\pi^2} \approx 0 \cdotp 30396 35509 \ldots$