An invitation to modern number theory
 Author/Creator
 Miller, Steven J., 1974
 Language
 English.
 Imprint
 Princeton : Princeton University Press, c2006.
 Physical description
 xx, 503 p. : ill ; 25 cm.
Access
Available online

Stacks

Unavailable
QA241 .M5344 2006
Assumed lost

Unavailable
QA241 .M5344 2006
More options
Contributors
 Contributor
 TaklooBighash, Ramin.
Contents/Summary
 Bibliography
 Includes bibliographical references (p. [476]496) and index.
 Contents

 PART 1. BASIC NUMBER THEORY
 1. Mod p Arithmetic, Group Theory and Cryptography
 2. Arithmetic Functions
 3. Zeta and LFunctions
 4. Solutions to Diophantine Equations
 PART 2. CONTINUED FRACTIONS AND APPROXIMATIONS
 5. Algebraic and Transcendental Numbers
 6. The Proof of Roth's Theorem
 7. Introduction to Continued Fractions
 PART 3. PROBABILISTIC METHODS AND EQUIDISTRIBUTION
 8. Introduction to Probability
 9. Applications of Probability: Benford's Law and Hypothesis Testing
 10. Distribution of Digits of Continued Fractions
 11. Introduction to Fourier Analysis
 12. f n k g and Poissonian Behavior
 PART 4. THE CIRCLE METHOD
 13. Introduction to the Circle Method
 14. Circle Method: Heuristics for Germain Primes
 PART 5. RANDOM MATRIX THEORY AND LFUNCTIONS
 15. From Nuclear Physics to LFunctions
 16. Random Matrix Theory: Eigenvalue Densities
 17. Random Matrix Theory: Spacings between Adjacent Eigenvalues
 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.
 Summary
 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 TaklooBighash 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.
 Supplemental links
 Table of contents
Subjects
Bibliographic information
 Publication date
 2006
 Responsibility
 Steven J. Miller and Ramin TaklooBighash.
 ISBN
 0691120609
 9780691120607