Miller, Steven J., Takloo–bighash, Ramin 　著

内容

目次

Foreword xi Preface xiii Notation xix PART 1. BASIC NUMBER THEORY1 Chapter 1. Mod p Arithmetic, Group Theory and Cryptography 3 Chapter2. Arithmetic Functions 29 Chapter 3. Zeta and L-Functions 47 Chapter 4.Solutions to Diophantine Equations 81 PART 2. CONTINUED FRACTIONS ANDAPPROXIMATIONS 107 Chapter 5. Algebraic and Transcendental Numbers 109Chapter 6. The Proof of Roth's Theorem 137 Chapter 7. Introduction toContinued Fractions 158 PART 3. PROBABILISTIC METHODS ANDEQUIDISTRIBUTION 189 Chapter 8. Introduction to Probability 191 Chapter9. Applications of Probability: Benford's Law and Hypothesis Testing 216Chapter 10. Distribution of Digits of Continued Fractions 231 Chapter 11.Introduction to Fourier Analysis 255 Chapter 12. f n k g and PoissonianBehavior 278 PART 4. THE CIRCLE METHOD 301 Chapter 13. Introductionto the Circle Method 303 Chapter 14. Circle Method: Heuristics for GermainPrimes 326 PART 5. RANDOM MATRIX THEORY AND L-FUNCTIONS 357 Chapter15. From Nuclear Physics to L-Functions 359 Chapter 16. Random MatrixTheory: Eigenvalue Densities 391 Chapter 17. Random Matrix Theory: Spacingsbetween Adjacent Eigenvalues 405 Chapter 18. The Explicit Formula andDensity Conjectures 421 Appendix A. Analysis Review 439 Appendix B.Linear Algebra Review 455 Appendix C. Hints and Remarks on the Exercises 463Appendix D. Concluding Remarks 475 Bibliography 476 Index 497