nt.number theory - Upper bound on Brun's constant

This seems to me like a simple matter of enumerating all the small twin primes, and then estimating the resulting error using a sieve bound. In particular, if $pi_2(x)$ is the number of twin primes $leq x$ then we know $pi_2(x) ll x (log x)^{-2}$. By a simple summation-by-parts exercise, this gives

$B=sum_{p,twin,p < X}frac{1}{p}+frac{1}{p+2}+O(log{X}^{-1})$.

I'm not sure what the numerical constant in the O-term is, but presumably it can be computed.