Prove that $ \lim_{N\to \infty} \sum_{n=1}^N \frac{1}{\phi(n)} $ exists or does not exist

Question

Prove that the limit exists or does not exist:

$$ \lim_{N\to \infty} \sum_{n=1}^N \frac{1}{\phi(n)}, $$

where $\phi(n)$ is the Euler Totient function.

The ratio test was inconclusive.

I'm fairly sure the p-series test says this series diverges because $p=1$ but then again in this case I'm not sure how to deal with a function in the place where $n$ normally is.

Answer 1

Since $\varphi(n)\leq n$ it follows that $$\frac{1}{\varphi(n)}\geq \frac{1}{n}$$ and hence $\sum \frac{1}{\varphi(n)}=\infty$.

Answer 2

A way of proof is the following:

$\dfrac{1}{p-1}\gt \dfrac1p$ and $\phi(p)=p-1$ then $$\sum_{n=1}^N \dfrac{1}{\phi(n)}\gt\sum_{p\text { prime}}^N \dfrac{1}{\phi(p)}\gt\sum_{{p\text { prime}}}^N \dfrac{1}{p}\to \infty$$ (because of the Euler's proof of the infinity of primes).