Brun's theorem

Statement

The theorem states the following:

For any $x>0$, the number of twin primes less than or equal to $x$ is bounded from above by:

${\frac {cx(\log \log x)^{2}}{(\log x)^{2}}}$

where $c$ is a positive constant independent of $x$.

As a corollary, we obtain that the sum, over all pairs of twin primes, of the reciprocals of both members of the pair, is finite. Specifically, the following sum is finite:

$\left({\frac {1}{3}}+{\frac {1}{5}}\right)+\left({\frac {1}{5}}+{\frac {1}{7}}\right)+\left({\frac {1}{11}}+{\frac {1}{13}}\right)+\dots$

The constant to which this sum converges is termed Brun's constant. No explicit finite upper bound on Brun's constant has been established.

The following are some easy corollaries:

1. The sum of the reciprocals of all twin primes is finite. Note that this sum differs from the sum above by $1/5$ since $1/5$ appears twice in the above sum. In other words, the set of twin primes is a small set.
2. The sum of the reciprocals of each of the lesser of the pairs of twin primes is finite.
3. The sum of the reciprocals of each of the greater of the pairs of twin primes is finite.