Selberg sieve

Summary

The Selberg sieve is an upper bound sieve of Atle Selberg. It replaces the indicator of "no prime factor below $z$" by the square of a sum of real weights. It then chooses the weights to minimise the main term, which is a quadratic form. It gives sharp upper bounds with a simple remainder term, for example the Brun–Titchmarsh bound for primes in arithmetic progressions. It is also called the $\Lambda^2$ sieve.