Question

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 $C$?

Answer 1

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.