publication . Thesis . 2005

# On Montgomery's pair correlation conjecture to the zeros of Riedmann zeta function

Li, Pei;
Open Access English
• Published: 01 Jan 2005
Abstract
In this thesis, we are interested in Montgomery's pair correlation conjecture which is about the distribution of.the spacings between consecutive zeros of the Riemann Zeta function. Our goal is to explain and study Montgomery's pair correlation conjecture and discuss its connection with the random matrix theory. In Chapter One, we will explain how to define the Ftiemann Zeta function by using the analytic continuation. After this, several classical properties of the Ftiemann Zeta function will be discussed. In Chapter Two, We will explain the proof of Montgomery's main result and discuss the pair correlation conjecture in detail. The main result about the pair c...
Subjects
arXiv: Mathematics::Number Theory
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1 Introduction of the Riemann Zeta function . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1.2 The Definition of the Riemann Zeta Function . . . . . . . . . . . . . . . . . . 2 1.3 The Ftiemann Zeta Function in the Region a > 1 and the Euler Product Formula 3 1.4 Analytic Continuation of C(s) to a > 0 . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Analytic Continuation of C(s) to @ and Functional Equation . . . . . . . . . 7 1.6 Zeros of the Ftiemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 Zero-free Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.8 Counting the Zeros of [(s) Inside a Rectangle . . . . . . . . . . . . . . . . . . 17 1.9 How to Find a Zero of Riemann Zeta Function . . . . . . . . . . . . . . . . . 21

2 The Pair Correlation of the Zeros of the Ftiemann Zeta Function . . . . . . . . . . . 2.1 The Pair Correlation of the Zeros of the Riemann Zeta Function . . . . . . . 2.1.1 Montgomery's Pair Correlation Conjecture . . . . . . . . . . . . . . 2.2 The Pair Correlation of Eigenvalues of The Gaussian Unitary Ensemble . . . 2.2.1 Random Matrix and The Gaussian Unitary Ensemble . . . . . . . . 2.2.2 Oscillator Wave Function . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Pair Correlation Function for the GUE . . . . . . . . . . . . . . . . 2.3 Numerical Support for Montgomery's Pair Correlation Conjecture and further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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