4 added 7 characters in body

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/13510/question-concerning-the-arithmetic-average-of-the-euler-phi-function

http://mathoverflow.net/questions/84571/averages-of-euler-phi-function-and-similar

3 added 83 characters in body

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 question: http://mathoverflow.net/questions/13510/question-concerning-the-arithmetic-average-of-the-euler-phi-function http://mathoverflow.net/questions/84571/averages-of-euler-phi-function-and-similar

2 added 224 characters in body

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? 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 question: http://mathoverflow.net/questions/13510/question-concerning-the-arithmetic-average-of-the-euler-phi-function http://mathoverflow.net/questions/84571/averages-of-euler-phi-function-and-similar

1