Is $a_n=\frac{1}{n}\sum_{k=1}^n\frac{\varphi(k)}{k}$ convergent? Let $(a_n)_{n\in\mathbb{N}}$ be defined as $a_n=\frac{1}{n}\sum_{k=1}^n\frac{\varphi(k)}{k}$ where $\varphi$ is the euler totient function. Is $(a_n)$ convergent. If so, what is its limit?
I have checked it numerically; it seems to converge to the value
$$
a\approx 0.6079384135652464404586775568799731914204890312331725
$$
However, I cannot think of a way to prove it.

See a result from this paper or the sum in the wiki article.  The sum may be approximated:
$$\sum_{k=1}^n \frac{\varphi(k)}{k} = \frac{6}{\pi^2}n + \mathcal{O}\left((\log n)^{2/3}(\log \log n)^{4/3}\right)$$
Since, as $n\to \infty$, $n>(\log n)^{2/3}(\log \log n)^{4/3}$, the big-O term drops out.  Thus, we're left with the limit being $\frac{6}{\pi^2}$.
