Given an integer $n$ and let $1\leq m\leq n$ be such that $n$ and $m$ are coprimes define $$ \mathcal{N_{n,m}}:=\text{the set of primes $p$ such that $p\equiv{m}\hspace{0.1cm}\mathrm{mod}(n)$}. $$ Let $\mathcal{P}$ be the set of all primes. I seem to recall that the following result is true: $$ \varphi(n)^{-1}=\lim_{k\to\infty}{\frac{|\mathcal{N_{n,m}}\cap\{1,2,\ldots,k\}|}{|\mathcal{P}\cap\{1,2,\ldots,k\}|}}. $$ where $\varphi$ is the Euler's function.

My question is two fold:

- Does anyone have a reference for the previous fact? I was unsuccesful finding it.
- Are there finer results along these lines? Second order results?

Thanks!