Sum of reciprocal of squares of all prime numbers
 

It is known that both the [URL="http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29"]armonic series [/URL] and the [URL="http://en.wikipedia.org/wiki/Proof_that_the_sum_of_the_reciprocals_of_the_primes_diverges"]sum of the reciprocals of the prime numbers[/URL] diverges
Informally written:
[TEX]\sum_p \frac{1}{p} = \infty[/TEX]

Considering that the sum of the reciprocal of all natural numbers converges, as seen in [URL="http://en.wikipedia.org/wiki/Basel_problem"]Basel problem[/URL], that is
[TEX]\sum_n \frac{1}{n^2} = \frac{\pi^2}{6}[/TEX]

I was wondering if the sum of the reciprocals of the squares of prime numbers converges, and if so to what number, that is
[TEX]\sum_p \frac{1}{p^2} = \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \frac{1}{11^2} + \ldots[/TEX]

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);[/CODE]
and it seems to converge to some number that starts with
0.45224661779206...

Any help is welcomed, thanks.

It is easily seen to be convergent by a comparison test:
[url]http://mathworld.wolfram.com/ComparisonTest.html[/url]

You are asking for the value of the [B]prime zeta function[/B] at 2:
[url]http://en.wikipedia.org/wiki/Prime_zeta_function[/url]
[url]http://mathworld.wolfram.com/PrimeZetaFunction.html[/url]

Wolfram gives references that may be helpful.

[quote=Damian;215999]the sum of the reciprocal of all natural numbers converges[/quote]You meant, "the sum of the reciprocals [I]of the squares[/I] of all natural numbers converges". :-)

Hi philmoore, thanks for your fast answer!
I wasn't aware of the prime zeta function. So yes, my question was basically about [TEX]P(2)[/TEX].
In the references it says that the firsts digits are
[TEX]P(2) \simeq 0.452247420041065498506543364832247934173231343239892421736418 \ldots[/TEX]
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.