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 answer 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.

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