An Invitation to Modern Number Theory

Erscheinungsjahr 2006
ISBN: 978-0-691-12060-7
Verlag: Princeton University Press

Buch, Englisch, 519 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 945 g

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.

Autoren/Hrsg.

Miller, Steven J.

Takloo-Bighash, Ramin

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

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

Über Autor(innen)

Steven J. Miller is an Assistant Professor of Mathematics at Brown University. Ramin Takloo-Bighash is an Assistant Professor of Mathematics at Princeton University.