It is a fact (as explained here) that for all $\varepsilon>0$ there is a natural number $n$ such that $\phi(n)/n < \varepsilon$, where $\phi$ is the Euler totient function.
I noticed that $$ \text{res}_{s=1} L(s, 1_n) = \lim_{s\to 1} (s-1)\zeta(s)\prod_{p:\ p|n} (1-1/p^s) = \prod_{p:\ p|n} (1-1/p) = \phi(n)/n $$ where $1_n$ is the principal Dirichlet character modulo $n$.
I was wondering if this observation can lead to a proof of the above fact.
Even though there is an elementary proof of the fact quoted above, there may be some insight to gain from a non-elementary solution and this is why I ask this question. Also, this gives me a chance to appreciate some concepts in analytic number theory, a subject I am new to.