An Invitation to Modern Number Theory. Steven J. Miller and Ramin Takloo-Bighash PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Transcription

1 An Invitation to Modern Number Theory Steven J. Miller and Ramin Takloo-Bighash PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

2 Contents Foreword Preface Notation xi xiii xix PART 1. BASIC NUMBER THEORY 1 Chapter 1. Mod p Arithmetic, Group Theory and Cryptography Cryptography Efficient Algorithms Clock Arithmetic: Arithmetic Modulo n Group Theory RSA Revisited Eisenstein's Proof of Quadratic Reciprocity 21 Chapter 2. Arithmetic Functions Arithmetic Functions Average Order Counting the Number of Primes 38 Chapter 3. Zeta and L-Functions The Riemann Zeta Function Zeros of the Riemann Zeta Function Dirichlet Characters and.l-functions 69 Chapter 4. Solutions to Diophantine Equations Diophantine Equations Elliptic Curves Height Functions and Diophantine Equations Counting Solutions of Congruences Modulo p Research Projects 105 PART 2. CONTINUED FRACTIONS AND APPROXIMATIONS 107 Chapter 5. Algebraic and Transcendental Numbers 109

3 Viii 5.1 Russell's Paradox and the Banach-Tarski Paradox Definitions Countable and Uncountable Sets Properties of e Exponent (or Order) of Approximation Liouville's Theorem Roth's Theorem 132 Chapter 6. The Proof of Roth's Theorem Liouville's Theorem and Roth's Theorem Equivalent Formulation of Roth's Theorem Roth's Main Lemma Preliminaries to Proving Roth's Lemma Proof of Roth's Lemma 155 Chapter 7. Introduction to Continued Fractions Decimal Expansions Definition of Continued Fractions Representation of Numbers by Continued Fractions Infinite Continued Fractions Positive Simple Convergents and Convergence Periodic Continued Fractions and Quadratic Irrationals Computing Algebraic Numbers'Continued Fractions Famous Continued Fraction Expansions Continued Fractions and Approximations Research Projects 186 PART 3. PROBABILISTIC METHODS AND EQUIDISTRIBUTION 189 Chapter 8. Introduction to Probability Probabilities of Discrete Events Standard Distributions Random Sampling The Central Limit Theorem 213 Chapter 9. Applications of Probability: Benford's Law and Hypothesis Testing Benford's Law Benford's Law and Equidistributed Sequences Recurrence Relations and Benford's Law Random Walks and Benford's Law Statistical Inference Summary 229 Chapter 10. Distribution of Digits of Continued Fractions Simple Results on Distribution of Digits Measure of a with Specified Digits 235

4 IX 10.3 The Gauss-Kuzmin Theorem Dependencies of Digits Gauss-Kuzmin Experiments Research Projects 252 Chapter 11. Introduction to Fourier Analysis Inner Product of Functions Fourier Series Convergence of Fourier Series Applications of the Fourier Transform Central Limit Theorem Advanced Topics 276 Chapter 12. {n k a} and Poissonian Behavior Definitions and Problems Densenessof {n k a} Equidistribution of {n k a} Spacing Preliminaries Point Masses and Induced Probability Measures Neighbor Spacings Poissonian Behavior Neighbor Spacings of {n k a} Research Projects 299 PART 4. THE CIRCLE METHOD 301 Chapter 13. Introduction to the Circle Method Origins The Circle Method Goldbach's Conjecture Revisited 315 Chapter 14. Circle Method: Heuristics for Germain Primes Germain Primes Preliminaries The Functions FN {X) and u(x) Approximating FN (X) on the Major Arcs Integrals over the Major Arcs Major Arcs and the Singular Series Number of Germain Primes and Weighted Sums Exercises Research Projects 354 PART 5. RANDOM MATRIX THEORY AND L-FUNCTIONS 357 Chapter 15. From Nuclear Physics to L-Functions Historical Introduction Eigenvalue Preliminaries 364

5 X 15.3 Semi-Circle Law Adjacent Neighbor Spacings Thin Sub-families Number Theory Similarities between Random Matrix Theory and L-Functions Suggestions for Further Reading 390 Chapter 16. Random Matrix Theory: Eigenvalue Densities Semi-Circle Law Non-Semi-Circle Behavior Sparse Matrices Research Projects 403 Chapter 17. Random Matrix Theory: Spacings between Adjacent Eigenvalues Introduction to the 2 x 2 GOE Model Distribution of Eigenvalues of 2 x 2 GOE Model Generalization to N x N GOE Conjectures and Research Projects 418 Chapter 18. The Explicit Formula and Density Conjectures Explicit Formula Dirichlet Characters from a Prime Conductor Summary of Calculations 437 Appendix A. Analysis Review 439 A.I Proofs by Induction 439 A.2 Calculus Review 442 A.3 Convergence and Continuity 447 A.4 Dirichlet's Pigeon-Hole Principle 448 A.5 Measures and Length 450 A. 6 Inequalities 452 Appendix B. Linear Algebra Review 455 B.I Definitions 455 B.2 Change of Basis 456 B.3 Orthogonal and Unitary Matrices 457 B.4 Trace 458 B.5 Spectral Theorem for Real Symmetric Matrices 459 Appendix C. Hints and Remarks on the Exercises 463 Appendix D. Concluding Remarks 475 Bibliography 476 Index 497