Riemann Zero spacings and Montgomery's pair correlation conjecture

This paper presents an overview of mathematical work surrounding Montgomery’s pair correlation conjecture. The first chapter introduces the Riemann zeta function and Riemann’s method of computation of the first several zeros on the vertical line 1/2 + it. Chapter 2 presents Montgomery’s pair correlation conjecture following his original paper from 1971. Chapter 3 concerns the Gaussian Unitary Ensemble of random matrices, used to model particle physics and having eigenvalue distribution paralleling the distribution of nontrivial zeros of the Riemann zeta function, as well as touching on similar matrix ensembles. Chapter 4 presents empirical results of the distribution of nontrivial zeros, obtained computationally by Odlyzko, and the methods used to obtain them. The final chapter presents brief highlights of recent results which contribute to the growing body of Riemann zeta-to-physics and Riemann zeta-to-random matrix theory correspondence.