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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!

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This is due to Dirichlet. See en.wikipedia.org/wiki/… –  wood Apr 6 '11 at 12:59
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@wood: Clearly not, it implies the PNT. –  Charles Apr 6 '11 at 14:29
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@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. –  Gerry Myerson Apr 7 '11 at 0:07
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2 Answers

up vote 10 down vote accepted

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).

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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. –  Gerry Myerson Apr 6 '11 at 13:19
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+1 for the pointer to Siegel-Walfisz. –  Gerry Myerson Apr 6 '11 at 13:21
    
I don't have the reputation to comment but what about the second question? Are there more precise results? –  Val Apr 6 '11 at 13:27
    
@Gerry Myerson, of course you are right I was imprecise (and will change this). And, sorry for the wrong spelling of your name. –  quid Apr 6 '11 at 13:31
    
Yes, see 2nd paragraph of unknown (google)'s answer for keywords. –  Gerry Myerson Apr 6 '11 at 13:31
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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.

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