A Primer of Analytic Number Theory: From Pythagoras to Riemann

by Jeffrey Stopple

• ISBN13: 9780521813099
• ISBN10: 0521813093
• Format: Hardcover
• Copyright: 2003-06-23
• Publisher: Cambridge University Press

Summary

This undergraduate introduction to analytic number theory develops analytic skills in the course of studying ancient questions on polygonal numbers, perfect numbers and amicable pairs. The question of how the primes are distributed amongst all the integers is central in analytic number theory. This distribution is determined by the Riemann zeta function, and Riemann's work shows how it is connected to the zeroes of his function, and the significance of the Riemann Hypothesis. Starting from a traditional calculus course and assuming no complex analysis, the author develops the basic ideas of elementary number theory. The text is supplemented by series of exercises to further develop the concepts, and includes brief sketches of more advanced ideas, to present contemporary research problems at a level suitable for undergraduates. In addition to proofs, both rigorous and heuristic, the book includes extensive graphics and tables to make analytic concepts as concrete as possible.

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. Pelf'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