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

2$\begingroup$ Naive question: why does one expect the value of this sum to have interesting properties? $\endgroup$ – Yemon Choi Jan 27 '11 at 3:47

5$\begingroup$ 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
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 $$N=\sum_p{1\over p^2}=\sum_{k=1}^{\infty}{\mu(k)\over k}\log(\zeta(2k))=0.4522474200\dots$$

40$\begingroup$ suspiciously close to the lb/kg ratio.... $\endgroup$ – Yaakov Baruch Jan 27 '11 at 4:18

1$\begingroup$ An older source is in Euler's Introductio in analysin infinitorum, volume 1 chapter 15 (1748) eulerarchive.maa.org/pages/E101.html . An English translation is available here 17centurymaths.com/contents/introductiontoanalysisvol1.htm $\endgroup$ – Frédéric Grosshans Jun 11 '16 at 9:50

1$\begingroup$ In particular, on page 480 of 17centurymaths.com/contents/euler/introductiontoanalysisvolone/… $\endgroup$ – Gerry Myerson Jun 11 '16 at 14:02

$\begingroup$ hahaha, @YaakovBaruch nice; $4/(\pi^2 1)$ is close too hahaha $\endgroup$ – carlosayam Apr 12 '17 at 20:43