Online

Dalton, Jack Robert

Mathematics

2017

MS

A number of exciting recent developments in the field of sieve theory have been done concerning bounded gaps between prime numbers. One of the main techniques used in these papers is a modified version of Selberg's Sieve from the 1940's. While there are a number of sources that explain the original sieve, most, if not all, are quite inaccessible to those without significant experience in analytic number theory. The goal of this exposition is to change that. The statement and proof of the general form of Selberg's sieve is, by itself, difficult to understand and appreciate. For this reason, the inital exposition herein will be about one particular application: to recover Chebysheff's upper bound on the order of magnitude of the number of primes less than a given number. As Selberg's sieve follows some of the same initial steps as the more elementary sieve of Eratosthenes, this latter sieve will be worked through as well. To help the reader get a better sense of Selberg's sieve, a few particular applications are worked through, including an upper bound on the number of twin primes less than a number. This will then be used to show the convergence of the reciprocals of the twin primes.