An Invitation to Modern Number Theory (Hardcover) (書側有些許黴斑,不影響閱讀 不介意再下單)

Description

In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research.

Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory.

Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class.

 

Table Of Contents

Foreword xi
Preface xiii
Notation xix

PART 1. BASIC NUMBER THEORY 1

Chapter 1. Mod p Arithmetic, Group Theory and Cryptography 3
Chapter 2. Arithmetic Functions 29
Chapter 3. Zeta and L-Functions 47
Chapter 4. Solutions to Diophantine Equations 81

PART 2. CONTINUED FRACTIONS AND APPROXIMATIONS 107

Chapter 5. Algebraic and Transcendental Numbers 109
Chapter 6. The Proof of Roth's Theorem 137
Chapter 7. Introduction to Continued Fractions 158

PART 3. PROBABILISTIC METHODS AND EQUIDISTRIBUTION 189

Chapter 8. Introduction to Probability 191
Chapter 9. Applications of Probability: Benford's Law and Hypothesis Testing 216
Chapter 10. Distribution of Digits of Continued Fractions 231
Chapter 11. Introduction to Fourier Analysis 255
Chapter 12. f n k g and Poissonian Behavior 278

PART 4. THE CIRCLE METHOD 301

Chapter 13. Introduction to the Circle Method 303
Chapter 14. Circle Method: Heuristics for Germain Primes 326

PART 5. RANDOM MATRIX THEORY AND L-FUNCTIONS 357

Chapter 15. From Nuclear Physics to L-Functions 359
Chapter 16. Random Matrix Theory: Eigenvalue Densities 391
Chapter 17. Random Matrix Theory: Spacings between Adjacent Eigenvalues 405
Chapter 18. The Explicit Formula and Density Conjectures 421

Appendix A. Analysis Review 439
Appendix B. Linear Algebra Review 455
Appendix C. Hints and Remarks on the Exercises 463
Appendix D. Concluding Remarks 475

Bibliography 476
Index 497