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?
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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. 


you can also see this : numbers.computation.free.fr/Constants/Primes/… 

