Sieve Methods Lecture Notes, Part I the Brun-hooley Sieve

A sieve is a technique for bounding the size of a set after the elements with “undesirable properties” (usually of a number theoretic nature) have been removed. The undesirable properties could be divisibility by a prime from a given set, other multiplicative constraints (divisibility by a perfect square for example) or inclusion in a set of residue classes. The methods usually involve some kind of combinatorial reasoning where one “sees” the removal process going on in some form or another. Sieve methods have also been used in the literature to describe procedures for attacking problems about detecting primes in sequences of integers, whether or not there is any kind of “removal process” going on or not. The original sieve is, of course, the Sieve of Eratosthenes, the familiar process of creating a table of prime numbers by systematically removing those integers divisible by small primes (but keeping the primes themselves). The modern sieve was created by Viggo Brun in the period 1915-1922 as a way of attacking famous unsolved problems such as Golbach’s Conjecture and the Twin Prime problem (both, so far, unsuccessfully). Sieve methods have since found enormous application in number theory, in particular to such problems as • studying k-tuples of primes in special configurations (generalizations of twin primes); • studying primes in short intervals; • studying primes in arithmetic progressions; • analyzing the multiplicative structure of integers (distribution of prime factors and divisors); • analyzing the structure of shifted primes p−1, p+1with application to the distribution of arithmetic functions such as φ(n) and σ(n); • studying the distribution of sizes of circles in Apollonian circle packings.