An Invitation to Modern Number Theory

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

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

Review :
"This is a great book... [I]t is a fine book for talented and mathematically mature undergraduates, for graduate students, and for anyone looking for information on modern number theory."--Henry Ricardo, MAA Reviews "This is the first text to present Random Matrix Theory and the Circle Method for German primes. This well-written book supplements classic texts by showing connections between seemingly diverse topics, by making the subject accessible to beginning students and by whetting their appetite for continuing in mathematics"--Mathematical Reviews "I would highly recommend this book to anybody interested in number theory, from an undergraduate student to an established expert, since everybody will be able to find in this book lots of new interesting material, tempting problems, and interesting computational challenges. It could also be used as a textbook for a graduate course in number theory. To promote and stimulate independent research, it contains many very interesting exercises and even suggestions for research projects."--Igor Shparlinski, SIAM Review