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.

  • $\begingroup$ Sorry, I just want to check for clarity, is your question (more simply stated): "prove that limit exists"? $\endgroup$
    – Ben Weiss
    Dec 17 '09 at 1:23
  • $\begingroup$ ``Clearly'' the analytic density here is log_10 2, by Benford's law. $\endgroup$ Dec 17 '09 at 2:04
  • $\begingroup$ @Michael: I don't understand why Benford's law "clearly" applies here. I understand the principle of it is at work, but I don't think it is so obvious you can apply it. As Gjergji explains below you first must do some work to show equidistribution. $\endgroup$
    – Ben Weiss
    Dec 17 '09 at 2:08
  • $\begingroup$ Ben: the quotes around the first word of my previous comment were meant to indicate sarcasm. $\endgroup$ Dec 17 '09 at 2:11
  • 2
    $\begingroup$ my mistake...I failed the class in sarcasm. It's why they kicked me out of NYC and I'm now in the midwest. $\endgroup$
    – Ben Weiss
    Dec 17 '09 at 2:14

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.

  • 4
    $\begingroup$ The references Ben gave contain the answer and much more. The main idea here is to show that the log of you values become equidistributed $\pmod{1}$. The Benford behavior then follows by exponentiation. $\endgroup$ Dec 17 '09 at 1:55
  • $\begingroup$ Great - thanks. I don't have easy access to these papers but I'll try to dig them up! $\endgroup$
    – Alon Amit
    Dec 17 '09 at 6:52

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