Sieve methods, old and new

The number-theoretic sieve as a tool for studying primes is of very ancient lineage dating from Eratosthenes, nevertheless it remained almost unchanged through two millennia.

During the twentieth century there have been a number of periods of substantial progress. Although the main questions are still unanswered we can now begin to see cases where the sieve is finally able to successfully attack problems of the type for which it was designed.

In this lecture we discuss some of the history and basic ideas behind the main steps along this path.