A quantitative form of the twin prime conjecture asserts that the the number of twin primes less than $n$ is asymptotically equal to $2\, C\, n/ \ln^2(n)$ where $C$ is the socalled twin prime constant. A variety of sieve methods (originating with Brun) can be used show that the number of twin primes less than $n$ is at most $A\, n/ \ln^2 (n) $ for some constant $A>2C$. My question is: What is the smallest known value of $A$? I'd also be interested in learning what the best known constants are for the prime ktuple conjecture?
1 Answer
J Wu, Chen's double sieve, Goldbach's conjecture, and the twin prime problem, Acta Arith 114 (2004) 215273, MR 2005e:11128, bounds the number of twin primes above by $2aCx/\log^2x$, with $C=\prod p(p2)/(p1)^2$, and $a=3.3996$; I don't know whether there have been any improvements.

3$\begingroup$ I have checked Math Reviews for papers and reviews that cite Wu's paper. As far as I can tell from the reviews, there is no claim of an improvement on Wu's result. $\endgroup$ Aug 9, 2010 at 23:32