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 L-Functions
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 L-FUNCTIONS
15. From Nuclear Physics to L-Functions
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 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.