Texts and Readings in Mathematics

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.

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.
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
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
Bibliography
Index