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](http://en.wikipedia.org/wiki/Euler%27s_totient_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! 



Asymptotic Distribution of Primes
A good way to find the result you mentioned is to search for Dirichlet's (prime number) theorem; while Dirichlet only proved the infinitude of the set in question, nowadays one will frequently find the more precise assertion you mentioned when this result is discussed. 


A more common way to state it is that the number of primes congruent to $m$ modulo $n$ smaller than $x$ is asymptotically equal to $\varphi(n)^{-1} x/log (x) $ (assuming coprimeness as you did), which in combination with the prime number theorem implies what you are looking for.


There are a variety of results related to finer aspects of this problem;
key words e.g. Bombieri-Vinogradov Theorem or Siegel-Walfiz Theorem. 


See for example the wikipedia article on Dirichlet's theorem [here](http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions) which also links to the keywords I mentioned for a quick overview. 


Other than that as Gerry Myerson said any typical book on Analytic Number Theory will contain something on this subject (how much depends of course on the book).


