## Texts and Readings in Mathematics

An Invitation to Modern Number Theory

Steven J. Miller and Ramin Takloo-Bighash

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.

Reviews

"This is a great book. . . . [I]t is a fine book for talented and mathematically mature undergraduates, for graduate students, and for anyone looking for information on modern number theory."

Contents

PART 1. BASIC NUMBER THEORY1. 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

Reprint for sale in India only

Texts and Readings in Mathematics/ 61

2012, 9789380250373, 524 pages, paper cover, Rs. 790.00

Texts and Readings in Mathematics/ 61

2012, 9789380250373, 524 pages, paper cover, Rs. 790.00