The sum over primes involving powers of 10

[Klaus_Hoffmann]

recently i saw on a book (Apostol Analytic Number theory if i am not wrong) the prime calculating expression $$\sum_{p} 10^{-p}=S$$

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

$$S=0.2003000500007....$$

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?.

 

[Gib Z]

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 $0.0110101000101001$. And no, it wouldn't be as efficient as other methods such as the Prime Number Theorem, and that's not great either but that says that $p_n$~$n \log_e n$ where p_n denotes the nth prime number.

 

[robert Ihnot]

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.