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.

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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.

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An Invitation to Modern Number Theory

An Invitation to Modern Number Theory

An Invitation to Modern Number Theory

An Invitation to Modern Number Theory

Overview

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.

Product Details

ISBN-13: 9780691215976
Publisher: Princeton University Press
Publication date: 07/21/2020
Sold by: Barnes & Noble
Format: eBook
Pages: 528
File size: 3 MB

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.

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


What People are Saying About This

William Duke

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

From the Publisher

"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 Dueñez, University of Texas, San Antonio

Eduardo Duenez

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

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