An Invitation to Modern Number Theory

Description

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.

• Author: Steven J. Miller, Ramin Takloo-Bighash
• Publisher: Princeton University Press
• Published: 2006-03-26
• Pages: 532
• ISBN-13: 9780691120607