An Invitation to Modern Number Theory / / Ramin Takloo-Bighash, Steven J. Miller.

• Superior document: Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Backlist 2000-2013
• VerfasserIn: Miller, Steven J.,; Takloo-Bighash, Ramin,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2021]; ©2006
• Year of Publication: 2021
• Language: English
• Physical Description: 1 online resource (528 p.) :; 20 line illus.
• Other title: Frontmatter --; Contents --; Foreword --; Preface --; Notation --; PART 1. BASIC NUMBER THEORY --; Chapter 1. Mod p Arithmetic, Group Theory and Cryptography --; Chapter 2. Arithmetic Functions --; Chapter 3. Zeta and L-Functions --; Chapter 4. Solutions to Diophantine Equations --; PART 2. CONTINUED FRACTIONS AND APPROXIMATIONS --; Chapter 5. Algebraic and Transcendental Numbers --; Chapter 6. The Proof of Roth's Theorem --; Chapter 7. Introduction to Continued Fractions --; PART 3. PROBABILISTIC METHODS AND EQUIDISTRIBUTION --; Chapter 8. Introduction to Probability --; Chapter 9. Applications of Probability: Benford's Law and Hypothesis Testing --; Chapter 10. Distribution of Digits of Continued Fractions --; Chapter 11. Introduction to Fourier Analysis --; Chapter 12. {nk∝} and Poissonian Behavior --; PART 4. THE CIRCLE METHOD --; Chapter 13. Introduction to the Circle Method --; Chapter 14. Circle Method: Heuristics for Germain Primes --; PART 5. RANDOM MATRIX THEORY AND L-FUNCTIONS --; Chapter 15. From Nuclear Physics to L-Functions --; Chapter 16. Random Matrix Theory: Eigenvalue Densities --; Chapter 17. Random Matrix Theory: Spacings between Adjacent Eigenvalues --; Chapter 18. The Explicit Formula and Density Conjectures --; Appendix A. Analysis Review --; Appendix B. Linear Algebra Review --; Appendix C. Hints and Remarks on the Exercises --; Appendix D. Concluding Remarks --; Bibliography --; Index
• Format: Mode of access: Internet via World Wide Web.
• ISBN: 9780691215976; 9783110442502
• DOI: 10.1515/9780691215976?locatt=mode:legacy
• Access: restricted access
• Hierarchical level: Monograph
• Statement of Responsibility: Ramin Takloo-Bighash, Steven J. Miller.

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.