## Work down on the Upper bound of the Twin Primes

It can be shown using the Selberg Sieve method, that the maximum number of Twin primes less than $N$ is $$\frac{CN}{\ln^2(N)}$$ does anyone know if there has been any work done on finding an upper bound for the constant $C$?

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 Essentially the same question was asked here: mathoverflow.net/questions/34719/… – Mark Lewko Mar 15 2011 at 20:30

It is conjectured that the number of twin primes less than $N$ is $(\mathfrak{S}+o(1))N/(\log N)^2$, where $$\mathfrak{S}=2\prod_{p>2}(1-(p-1)^{-2})$$ is the so-called twin-prime constant. Using the large sieve it is easy to show that the number of twin primes less than $N$ is at most $(8\mathfrak{S}+o(1))N/(\log N)^2$. According to page 76 of Tenenbaum's Introduction to analytic and probabilistic number theory, the best result in this direction is by Wu (1990) which replaces 8 by 3.418.

EDIT: According to MathSciNet, Wu (2004) improved 3.418 to 3.3996.

EDIT: The constant 8 also follows from the Selberg sieve, see page 65 in Greaves' Sieves in number theory.

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And of course the result (for some $C$) is due to V. Brun. – Denis Chaperon de Lauzières Mar 15 2011 at 15:47
I think the Brun sieve misses a power of $\log\log N$. – GH Mar 15 2011 at 15:54
Only the first version of Brun's sieve has this $\log \log N$; he improved it later to get the right order of magnitude. – Denis Chaperon de Lauzières Mar 15 2011 at 17:35
Great, so Brun was the first to get the right order of magnitude. Do you know a reference? – GH Mar 15 2011 at 19:13
gallica.bnf.fr/ark:/12148/bpt6k3121p/… – Denis Chaperon de Lauzières Mar 15 2011 at 20:26
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