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
$\lim_{s \to 1^+} \frac{\sum_{p \in P} p^{-s}}{\log(\frac{1}{s-1})}$.
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).
How would one go about proving that P has an analytic density?
EDIT: This question has been asked again a few years ago, and it is my fault. I did accept Ben Weiss' answer but I couldn't back then check the papers, and it turns out that they don't actually answer my question! So, please refer to the newer version of the question for additional information.
Best Answer
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.)
The first (among other things) proves that the density of primes with leading coefficient $k$ is $\log_{10}\left(\frac{k +1}{k}\right).$
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)
The second is a more general statement about first digits. It is
The first digit problem by Ralph Raimi in American Math Monthly vol 83 No 7
Hope this all helps.