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The sum over primes involving powers of 10

  1. Jun 30, 2007 #1
    recently i saw on a book (Apostol Analytic Number theory if i am not wrong) the prime calculating expression [tex] \sum_{p} 10^{-p}=S [/tex]

    where the sum was extended to all the prime numbers, if i am right

    [tex] S=0.2003000500007............. [/tex]

    so knowing the value of 'S' you could get the primes, hence here is my questions if we knew an 'accelerating' process for the series above in a similar way to Euler transformation or Euler-Laurin sum formula, could we get an efficient algorithm to calculate primes?.
  2. jcsd
  3. Jul 1, 2007 #2

    Gib Z

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    Homework Helper

    Your value for S seems to be a bit troubled >.< The only digits one should see are 1's and 0's. I get around [itex]0.0110101000101001[/itex]. And no, it wouldn't be as efficient as other methods such as the Prime Number Theorem, and thats not great either but that says that [itex]p_n[/itex]~[itex] n \log_e n[/itex] where p_n denotes the nth prime number.
  4. Jul 20, 2007 #3
    Usually these things are calculated from the primes, not the other way around. Which is to say that S can only be calculated to the extent we have already determined the primes.
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