J Wu, Chen's double sieve, Goldbach's conjecture, and the twin prime problem, Acta Arith 114 (2004) 215-273, MR 2005e:11128, bounds the number of twin primes above by 2aCx/log2x, with C=prodp(p−2)/(p−1)2, and a=3.3996; I don't know whether there have been any improvements.
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