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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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marked as duplicate by Gerry Myerson, Andres Caicedo, Ramiro de la Vega, Daniel Moskovich, Andrey Rekalo Jul 9 '13 at 5:33

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Essentially the same question was asked here: mathoverflow.net/questions/34719/… –  Mark Lewko Mar 15 '11 at 20:30
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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 '11 at 15:47
    
I think the Brun sieve misses a power of $\log\log N$. –  GH from MO Mar 15 '11 at 15:54
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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 '11 at 17:35
    
Great, so Brun was the first to get the right order of magnitude. Do you know a reference? –  GH from MO Mar 15 '11 at 19:13
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