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The question if there is an upper bound known for Brun's constant was discussed briefly here: http://gowers.wordpress.com/2009/05/22/what-is-wolfram-alpha-good-for/ but no sure answer was given.

So I thought I'd ask the question here. Can one get any upper bound for the sum of the reciprocals of the twin primes?

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For nonexperts: note that Brun's constant is precisely the sum of the reciprocals of the twin primes. –  Pete L. Clark May 1 '10 at 21:51
    
I would say that en.wikipedia.org/wiki/Brun%27s_constant gives a nice account of the knowledge (including T Nicely's 2010 computation). As for David Hansen's answer, it seems that Cramér's model is needed for sharp estimates of remainders. –  Wadim Zudilin May 2 '10 at 8:15
    
What?! How is Cramer's model related at all? –  David Hansen May 2 '10 at 19:36

2 Answers 2

up vote 4 down vote accepted

Crandall and Pomerance, "Prime numbers: a computational perspective" (Google books) says that Brun's constant B, the sum of the reciprocals of the twin primes, is known to be between 1.82 and 2.15.

edited to add: I'm aware that this isn't much of a citation. It would be nice if someone who has access to this book could give a better citation. I'd do it, but I'm not near a library today.

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Great. So now, to disprove the twin prime conjecture, all I have to do is show that the sum of the reciprocals of the first N is exactly 2.15! –  Steven Gubkin May 1 '10 at 21:20
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The Crandall-Pomerance statement is on page 14 of the book. The result is credited to T Nicely, Enumeration to 1.6 x 10^15 of the twin primes and Brun's constant, 1999, lasi.lynchburg.edu/Nicely_T/Public/twins/twins2.htm –  Gerry Myerson May 2 '10 at 1:16
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I couldn't get Gerry's link to work, but this one might be relevant - trnicely.net/index.html –  François G. Dorais May 2 '10 at 4:28
    
One can find a limited preview of Pomerance and Crandall on google books. They give an upper bound of approximately 2.4 but do not credit Nicely with this result. Even without any reference to the literature I am willing to take them at their word. The answer is in fact better than I hoped, in the sense that the upper and lower bound are not ridiculously far apart. –  Johan May 2 '10 at 16:02

This seems to me like a simple matter of enumerating all the small twin primes, and then estimating the resulting error using a sieve bound. In particular, if $\pi_2(x)$ is the number of twin primes $\leq x$ then we know $\pi_2(x) \ll x (\log x)^{-2}$. By a simple summation-by-parts exercise, this gives

$B=\sum_{p\,twin,p < X}\frac{1}{p}+\frac{1}{p+2}+O(\log{X}^{-1})$.

I'm not sure what the numerical constant in the O-term is, but presumably it can be computed.

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