The number theory identity $\phi(1) + \phi(2) + \dots + \phi(n) \approx \frac{3n^2}{\pi^2}$ can be interpreted as counting relatively prime pairs of numbers $0 \leq \{ x,y \} \leq n$ .
Has anyone studied the distribution of error term? $\displaystyle \frac{1}{n} \left[\sum_{k=1}^n \phi(k) - \frac{3n^2}{\pi^2}\right]$ It looks like white noise:
The histogram has a distinctive shape, maybe hard to prove. I suspect it's the Gaussian Unitary Ensemble (a Hermite polynomial times a Gaussian).
Similar questionquestions:
http://mathoverflow.net/questions/84571/averages-of-euler-phi-function-and-similar
