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Does anyone know of any information/work on this sum? I found absolutely nothing on the web about it.

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    $\begingroup$ Naive question: why does one expect the value of this sum to have interesting properties? $\endgroup$
    – Yemon Choi
    Jan 27 '11 at 3:47
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    $\begingroup$ Ah, I see that someone more learned than me has provided links below, which presumably answer my question $\endgroup$
    – Yemon Choi
    Jan 27 '11 at 3:49
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    $\begingroup$ Kudos for not deleting "why does one expect the value of this sum to have interesting properties?" In retrospect, it was extremely degrading to Ethan Brush, wasn't it? - Well, he never came back to this site. $\endgroup$ Mar 8 at 18:16
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This would be $P(2)$, where $P$ is the "prime zeta function," q.v.

A couple of very old references are C. W. Merrifield, The Sums of the Series of Reciprocals of the Prime Numbers and of Their Powers, Proc. Roy. Soc. London 33 (1881) 4–10. doi:10.1098/rspl.1881.0063. JSTOR 113877

J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25 (1891) 347–362.

EDIT: A more recent source is Steven R Finch, Mathematical Constants, page 95: The sum of the squared reciprocals of primes is $$N=\sum_p{1\over p^2}=\sum_{k=1}^{\infty}{\mu(k)\over k}\log(\zeta(2k))=0.4522474200\dots$$

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