Analytic density of the set of primes starting with 1 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T13:45:16Z http://mathoverflow.net/feeds/question/9143 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9143/analytic-density-of-the-set-of-primes-starting-with-1 Analytic density of the set of primes starting with 1 Alon Amit 2009-12-17T01:19:40Z 2009-12-17T06:51:30Z <p>In 'Cours d'arithmetique', Serre mentions in passing the following fact (communicated to him by Bombieri): Let P be the set of primes whose first (most significant) digit in decimal notation is 1. Then P possesses an analytic density, defined as </p> <p><code>$\lim_{s \to 1^+} \frac{\sum_{p \in P} p^{-s}}{\log(\frac{1}{s-1})}$</code>.</p> <p>This is an interesting example since it's easy to see that this set does not have a 'natural' density, defined simply as the limit of the proportion of elements in P to the # of all primes up to $x$, as $x$ tends to infinity. Therefore the notion of analytic density is a genuine extension of the naive notion (they do coincide when both exist).</p> <p>How would one go about proving that P has an analytic density?</p> http://mathoverflow.net/questions/9143/analytic-density-of-the-set-of-primes-starting-with-1/9145#9145 Answer by Ben Weiss for Analytic density of the set of primes starting with 1 Ben Weiss 2009-12-17T01:37:40Z 2009-12-17T01:43:05Z <p>I think instead of posting my own explanation (which will only lose something in the translation) I'll instead refer you to two very interesting papers (thanks for posting this question, I haven't thought about this stuff in a couple years, and these papers were interesting reads to solve your problem.)</p> <p>The first (among other things) proves that the density of primes with leading coefficient $k$ is $\log_{10}\left(\frac{k +1}{k}\right).$</p> <p>Prime numbers and the first digit phenomenon by Daniel I. A. Cohen* and Talbot M. Katz in Journal of number theory 18, 261-268 (1984)</p> <p>The second is a more general statement about first digits. It is</p> <p>The first digit problem by Ralph Raimi in American Math Monthly vol 83 No 7</p> <p>Hope this all helps.</p>