Question

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?

Answer 1

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.