Notes for three seminars around the GPY sieve. The notes are based on the super nice article by Goldston in Matematik Dünyası as well as the Primes in tuples I paper itself. There are one or two inaccuracies in the notes (especially in the end where I confused 1/4 and 1/2...) and I deviate slightly when it comes to the contour integrals, but otherwise the notes contain a full proof of Bombieri-Davenport's result (that half the average gap appears infinitely often) and the GPY result that exponent of distribution >1/2 implies bounded gaps. For their limsup=0 result a calculation of Gallagher regarding the average of the singular series is needed, which I didn't do.
GPY seminars: gpy_seminars