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

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[ll]G. H. Hardy and J. Littlewood. Some problems inVparititionumerorum" iii: On the expression of a number as a sum of primes. Acta Math., 44:l-70, 1923. [OpenAIRE]

[12] Dennis A. Hejhal. On t h e triple correlation of zeros of the zeta function. Internat. Math. Res. Notices, (7):293ff., approx. 10 pp. (electronic), 1994.

[13] J . Keating. The Riemann zeta-function and quantum chaology. In Quantum chaos (Varenna, 1991), Proc. Internat. School of Phys. Enrico Fermi, CXIX, pages 145-185. North-Holland, Amsterdam, 1993.

[14] J. P. Keating and N. C. Snaith. Random matrices and L-functions. J. Phys. A, 36(12):2859- 2881, 2003. Random matrix theory.

[16] H. L. Montgomery. T h e pair correlation of zeros of the zeta function. In Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pages 181-193. Amer. Math. Soc., Providence, R.I., 1973.