2 - The Sieves of Brun and Selberg

Summary

Section 2.3 on Brun’s pure sieve gives some results which demonstrate its strength, for example, how the sieve can be used to derive both lower and upper bounds. Section 2.4 gives the estimation of Brun’s constant. Section 2.5 introduces a form of the Selberg sieve – it is variations of the sieve that have led to prime gap breakthroughs. An improved estimate is derived counting primes such that the shift by a given even integer is also prime. In Section 2.6 the Selberg sieve is used implicitly to prove the theorem of Bombieri and Davenport which gives a value in terms of the shift, for the leading terms. Section 2.7 has an application of the Selberg sieve to derive a weak form of the Brun–Titchmarsh inequality used later. Section 2.8 gives a description of 10 types of sieve with some key applications, e.g. the sieve of Eratosthenes, Legendre’s sieve, Brun’s combinatorial sieve, the large sieve, and Bombieri’s asymptotic sieve. This is followed by a reader’s guide to some of the main texts on sives. In Section 2.10 there is a discussion of the limits of sieve theory for capturing primes, the famous “parity problem”.