Asymptotic Distribution of Primes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:02:09Z http://mathoverflow.net/feeds/question/60812 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60812/asymptotic-distribution-of-primes Asymptotic Distribution of Primes ght 2011-04-06T12:56:10Z 2011-07-29T06:15:39Z <p>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 <a href="http://en.wikipedia.org/wiki/Euler%27s_totient_function" rel="nofollow">Euler's function</a>.</p> <p>My question is two fold: </p> <ul> <li>Does anyone have a reference for the previous fact? I was unsuccesful finding it.</li> <li>Are there finer results along these lines? Second order results?</li> </ul> <p>Thanks! </p> http://mathoverflow.net/questions/60812/asymptotic-distribution-of-primes/60813#60813 Answer by Gerry Myerson for Asymptotic Distribution of Primes Gerry Myerson 2011-04-06T12:59:48Z 2011-04-06T12:59:48Z <p>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. </p> http://mathoverflow.net/questions/60812/asymptotic-distribution-of-primes/60815#60815 Answer by quid for Asymptotic Distribution of Primes quid 2011-04-06T13:10:03Z 2011-04-06T13:39:17Z <p>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. </p> <p>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.</p> <p>There are a variety of results related to finer aspects of this problem; key words e.g. Bombieri-Vinogradov Theorem or Siegel-Walfiz Theorem. </p> <p>See for example the wikipedia article on Dirichlet's theorem <a href="http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" rel="nofollow">here</a> which also links to the keywords I mentioned for a quick overview. </p> <p>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).</p> http://mathoverflow.net/questions/60812/asymptotic-distribution-of-primes/60817#60817 Answer by Val for Asymptotic Distribution of Primes Val 2011-04-06T13:27:27Z 2011-04-06T13:27:27Z <p>I don't have the reputation to comment but what about the second question? Are there more precise results?</p>