Brun sieve

Summary

The Brun sieve is a combinatorial sieve method of Viggo Brun. It counts the integers of a set that have no prime factor below a bound $z$. It uses a truncated inclusion–exclusion, which gives upper and lower bounds with a controllable error. It was the first sieve to give nontrivial results on problems such as twin primes. There the sieve of Eratosthenes with full inclusion–exclusion produces too large an error. With it Brun proved that the sum of the reciprocals of the twin primes converges.