His best known contribution is that he showed that the sum of reciprocals of twin primes converges to a finite value, now called Brun's constant. He proved that by means of devising an improvement over Legendre's version the sieve of Eratosthenes, now known as the Brun sieve. Because of this, he is widely considered one of the founders of modern sieve theory.
- Brun's Constant: http://planetmath.org/encyclopedia/BrunsConstant.html
- Brun's Pure Sieve: http://planetmath.org/encyclopedia/BrunsPureSieve.html