Sifting the Primes
Abstract: These are the slides from a colloquium I presented at the University of Hawai'i on 18 March 2005. I begin by discussing the famous sieve of Eratosthenes, then demonstrate the power of sieve methods in tackling such outstanding problems as the Goldbach conjecture. In particular, I examine the weighted sieve of Chen Jing-Run, which he applied in 1974 to demonstrate that every sufficiently large even number N can be written in the form N = p + P2, where P2 denotes an integer with at most 2 prime factors.
I go on to discuss some of my recent work in which I proved a similar 'almost-primes' result concerning the values taken by pairs of irreducible binary quadratic forms. In the same sense that Chen's result is an approximation to the Goldbach conjecture, my own result is an approximation to Schinzel's celebrated hypothesis H.
Addendum: Since I presented the talk, I have been made aware of a paper of Diamond and Halberstam which extends the work of Halberstam and Richert to give a P7 result to the pairs of quadratics problem. Utilising their work, I reduced the number of prime factors in my problem from 6 to 5.