A Primer of Analytic Number Theory: From Pythagoras to Riemann

by Jeffrey Stopple

• ISBN13: 9780521012539
• ISBN10: 0521012538
• Format: Paperback
• Copyright: 2003-06-23
• Publisher: Cambridge University Press

Summary

This undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus. Jeffrey Stopple pays special attention to the rich history of the subject and ancient questions on polygonal numbers, perfect numbers and amicable pairs, as well as to the important open problems. The culmination of the book is a brief presentation of the Riemann zeta function, which determines the distribution of prime numbers, and of the significance of the Riemann Hypothesis.

Table of Contents

Preface	ix
Chapter 1. Sums and Differences	1	(23)
Chapter 2. Products and Divisibility	24	(19)
Chapter 3. Order of Magnitude	43	(21)
Chapter 4. Averages	64	(32)
Interlude 1. Calculus	83	(13)
Chapter 5. Primes	96	(50)
Interlude 2. Series	111	(35)
Chapter 6. Basel Problem	146	(13)
Chapter 7. Euler's Product	159	(34)
Interlude 3. Complex Numbers	187	(6)
Chapter 8. The Riemann Zeta Function	193	(23)
Chapter 9. Symmetry	216	(13)
Chapter 10. Explicit Formula	229	(31)
Interlude 4. Modular Arithmetic	254	(6)
Chapter 11. Pell's Equation	260	(14)
Chapter 12. Elliptic Curves	274	(21)
Chapter 13. Analytic Theory of Algebraic Numbers	295	(32)
Solutions	327	(48)
Bibliography	375	(4)
Index	379