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An Invitation to Modern Number Theory


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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

Promotional Information

The book provides a much-needed introduction to modern number theory that emphasizes analytic number theory. It should serve remarkably well as an advanced undergraduate textbook and its latter parts would be suitable for a beginning graduate course. Some of the material covered, such as the circle method and random matrix theory, is not readily available elsewhere in book form. These topics provide terrific examples of areas in number theory of great current interest that can be penetrated by students. I would seriously consider using this book in my own classes and recommend it with enthusiasm for highly motivated students. -- William Duke, University of California, Los Angeles Having this selection of material available in essentially self-contained form is fantastic. Reading the book (or taking a class based on it) might easily decide the future endeavors of many a neophyte mathematician. I have yet to discover a clearer exposition of the works of the circle method. The inclusion of exercises and, especially, of problems for further research and theoretical or numerical exploration is extremely valuable. I would dare to compare the book to Hardy and Wright's classic An Introduction to the Theory of Numbers in that Miller and Takloo-Bighash expose readers to the lively work of number theory, to its proofs, ideas, and methods, assuming only a very modest background. -- Eduardo Duenez, University of Texas, San Antonio

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.


"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

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