Brun’s Theorem and Sieve Methods

It is a well known fact that the sum of the reciprocal of primes diverges. This is one of many ways to show that there are infinitely many primes. Can we prove the infinitude of twin primes this way? Brun’s theorem gives a negative answer. But the proof of this negative result initiates the development of modern sieve theory, which is responsible for many recent positive results towards the conjecture. In this talk, we will give a proof of Brun’s theorem using Selberg’s Upper Bound Sieve.