Work down on the Upper bound of the Twin Primes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T12:08:19Z http://mathoverflow.net/feeds/question/58535 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58535/work-down-on-the-upper-bound-of-the-twin-primes Work down on the Upper bound of the Twin Primes Alex Botros 2011-03-15T14:25:47Z 2011-03-15T16:01:12Z <p>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 $C$?</p> http://mathoverflow.net/questions/58535/work-down-on-the-upper-bound-of-the-twin-primes/58542#58542 Answer by GH for Work down on the Upper bound of the Twin Primes GH 2011-03-15T15:45:28Z 2011-03-15T16:01:12Z <p>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.</p> <p>EDIT: According to MathSciNet, Wu (2004) improved 3.418 to 3.3996.</p> <p>EDIT: The constant 8 also follows from the Selberg sieve, see page 65 in Greaves' Sieves in number theory.</p>