Sieves

Sieve methods were created to attack the well-known Goldbach and twin-prime problems. Goldbach problem in strong form: is every reasonably large even number a sum of two prime numbers? Twin-prime problem: do pairs of consecutive odd numbers both prime, like 101 and 103, keep occurring for ever? It turns out that there are excellent reasons why sieve methods alone cannot solve these problems, but they give partial information on these and many other problems where the `deeper' methods of analytic number theory, such as exponential sums will not work. For example pairs of consecutive odd numbers which are either prime or very hard to factorise do keep on occurring.

Sieve methods can be purely combinatorial like the "sieve of Eratosthenes", or partly combinatorial and partly analytic: finding an inequality by rearranging a sum of squares which must be positive - even the modulus squared of an exponential sum in the so-called "large sieve".

References

• G. R. H. Greaves, Sieves in Number Theory, Springer 2001.
• C. Hooley, Applications of Sieve Methods, Cambridge 1976.
• G. R. H. Greaves et al. (ed.) Sieves, Exponential Sums, and their Applications in Number Theory , Cambridge 1997.