On Montgomery's pair correlation conjecture to the zeros of Riedmann zeta function
- Published: 01 Jan 2005
- Country: Canada
- Simon Fraser University Canada
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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