Anna Hall: The Zeta function and The Prime Number Theorem

Abstract:

The aim of this thesis is to investigate certain important properties of the Zeta function in the complex plane and use those to give a proof of The Prime Number Theorem. To do so we utilise the concept of analytic continuation, properties of the Gamma function, and briefly the Bernoulli numbers. In Chapter 1, we prove that the Gamma and Zeta functions are meromorphic in the complex plane, show different ways of representing the Zeta function, and end by investigating this function near the line Re(s)=1 in the complex plane. In Chapter 2, we utilise the results of Chapter 1 to look at the growth of the Zeta function near that same line, and together with Chebychev's Psi functions we prove two propositions and a theorem that together form the Prime Number Theorem. After the main chapters, we have Appendix A that covers the mathematical basis of analytic continuation and Appendix B which covers important concepts that are referred to in Chapters 1 and 2.