This exercise is meant to be 'explored' computationally. However, I implemented it in C++ and did not get anything better than a sequence of pseudo-random numbers.
Let $\Phi(n)=\sum_{i=1}^{n}\phi(n)$. Investigate the value of $\Phi(n)/n^2$ for increasingly large values of $n$, such as $n=100$, $n=1000$, and $n=10000$. Can you make a conjecture about the limit of this ratio as $n$ grows large without bound?
Notice that $\Phi(n)=n\phi(n)$. Hence, $\Phi(n)/n^2=\phi(n)/n$. Moreover, the largest value $\phi(n)/n$ ever attains is $1$ at $n=1$; everything else falls within the interval $(0,1)$, and the closest it gets to $1$ again is when $n$ is prime (since $\phi(p)=p-1$, and $(p-1)/p\approx1$ for very large primes $p$).
However, I am tempted to say that this function diverges, and that no conjecture about its limit can be concluded as a result.
What do you guys think?