5
$\begingroup$

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!

$\endgroup$
3
  • 1
    $\begingroup$ This is due to Dirichlet. See en.wikipedia.org/wiki/… $\endgroup$
    – wood
    Apr 6, 2011 at 12:59
  • 1
    $\begingroup$ @wood: Clearly not, it implies the PNT. $\endgroup$
    – Charles
    Apr 6, 2011 at 14:29
  • 3
    $\begingroup$ @Charles, in principle one could prove that the primes in one residue class are $1/\phi(n)$ of all the primes without knowing the asymptotics of either counting function, thus, without PNT, no? I agree that Dirichlet did not do this. $\endgroup$ Apr 7, 2011 at 0:07

2 Answers 2

10
$\begingroup$

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 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).

$\endgroup$
8
  • 1
    $\begingroup$ The $x/\log x$ result can't be Dirichlet's, as it would imply the prime number theorem, which came about 60 years after Dirichlet's work. $\endgroup$ Apr 6, 2011 at 13:19
  • 1
    $\begingroup$ +1 for the pointer to Siegel-Walfisz. $\endgroup$ Apr 6, 2011 at 13:21
  • $\begingroup$ I don't have the reputation to comment but what about the second question? Are there more precise results? $\endgroup$
    – Val
    Apr 6, 2011 at 13:27
  • $\begingroup$ @Gerry Myerson, of course you are right I was imprecise (and will change this). And, sorry for the wrong spelling of your name. $\endgroup$
    – user9072
    Apr 6, 2011 at 13:31
  • $\begingroup$ Yes, see 2nd paragraph of unknown (google)'s answer for keywords. $\endgroup$ Apr 6, 2011 at 13:31
5
$\begingroup$

It's just the prime number theorem for primes in arithmetic progression, no? Should be in any analytic number theory text that does the prime number theorem.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.