20100524, 22:38  #1 
May 2005
Argentina
272_{8} Posts 
Sum of reciprocal of squares of all prime numbers
It is known that both the armonic series and the sum of the reciprocals of the prime numbers diverges
Informally written: Considering that the sum of the reciprocal of all natural numbers converges, as seen in Basel problem, that is I was wondering if the sum of the reciprocals of the squares of prime numbers converges, and if so to what number, that is I tested numerically with Maxima software for the firsts primes with the code Code:
sum (if primep(x) then 1/x^2 else 0, x, 2, 100000); 0.45224661779206... Any help is welcomed, thanks. 
20100524, 22:53  #2 
"Phil"
Sep 2002
Tracktown, U.S.A.
2^{5}×5×7 Posts 
It is easily seen to be convergent by a comparison test:
http://mathworld.wolfram.com/ComparisonTest.html You are asking for the value of the prime zeta function at 2: http://en.wikipedia.org/wiki/Prime_zeta_function http://mathworld.wolfram.com/PrimeZetaFunction.html Wolfram gives references that may be helpful. 
20100524, 23:46  #3 
"Richard B. Woods"
Aug 2002
Wisconsin USA
7692_{10} Posts 

20100524, 23:57  #4 
May 2005
Argentina
2×3×31 Posts 
Hi philmoore, thanks for your fast answer!
I wasn't aware of the prime zeta function. So yes, my question was basically about . In the references it says that the firsts digits are and that P. Sebah found more than 10000 digits. So only the firsts 5 decimals where correct on my original post. (And I guess a closed form isn't known for this number) Thanks again for your very useful answer. 
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