3 corrected historical imprecision, and slight other edits

This fact goes typically by

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, a 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 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).

2 corrected spelling

This fact goes typically by Dirichlet's (prime number) theorem, 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).

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 here which also links to the keywords I mentioned for a quick overview.

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

1

This fact goes typically by Dirichlet's (prime number) theorem, 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).

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 here which also links to the keywords I mentioned for a quick overview.

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