Let $$\Phi_n=\frac{1}{n^2}\sum_{k=1}^n\varphi(k).$$ How one can show that $\Phi_n$ is convergent sequence? (Here, $\varphi$ denotes the Euler's totient function.) And please, without any monster asymptotic estimations.
Thank you!
EDIT:
Can we actually prove the convergence without finding the limit?
On
$\sum_{k=1}^n\varphi(k)$ counts pairs of integers $a,b$ with $a < b \le n$. So if we ignore the $a < b$ condition the thing we are taking the limit of is just the probability that if we take two numbers $a, b$ less than $n$ at random that they are relatively prime. For large enough $n$ we have the probability that two numbers is divisible by a prime $p$ is very close to $1/p^2$. Now doing an inclusion-exclusion over primes we get: $P(a,b \ \text{have a common factor}) = \sum_{p}p^{-2} - \sum_{p_1,p_2}p_1^{-2}p_2^{-2} + \sum_{p_1,p_2,p_3}p_1^{-2}p_2^{-2}p_3^{-2}-...$. So $P(a,b \text{ are relatively prime}) = (1-2^{-2})(1-3^{-2})... = 1/\zeta(2) = 6/ \pi^2$. Halving this to take into account the $a<b$ condition gives the desired result. Yea, I'm being a bit fuzzy on the convergence details but this is definitely the idea.
We certainly have $\varphi(k)\le k \space\forall k\ge 1$, and $\lim_{n\to \infty}\frac{1}{n^2}\sum_{k=1}^nk=\frac{1}{2}$. This shows that $\lim \sup \Phi_n\le 1/2$. It is known that $\Phi_n\sim \frac{3}{\pi^2}$.