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Prime Numbers and the Riemann Hypothesis

Prime numbers are beautiful, mysterious, and beguiling mathematical objects. The mathematician Bernhard Riemann made a celebrated conjecture about primes in 1859, the so-called Riemann hypothesis, which remains one of the most important unsolved problems in mathematics. Through the deep insights of the authors, this book introduces primes and explains the Riemann hypothesis. Students with a minimal mathematical background and scholars alike will enjoy this comprehensive discussion of primes. The first part of the book will inspire the curiosity of a general reader with an accessible explanation of the key ideas. The exposition of these ideas is generously illuminated by computational graphics that exhibit the key concepts and phenomena in enticing detail. Readers with more mathematical experience will then go deeper into the structure of primes and see how the Riemann hypothesis relates to Fourier analysis using the vocabulary of spectra. Readers with a strong mathematical background will be able to connect these ideas to historical formulations of the Riemann hypothesis.
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Table of Contents

1. Thoughts about numbers; 2. What are prime numbers?; 3. 'Named' prime numbers; 4. Sieves; 5. Questions about primes; 6. Further questions about primes; 7. How many primes are there?; 8. Prime numbers viewed from a distance; 9. Pure and applied mathematics; 10. A probabilistic 'first' guess; 11. What is a 'good approximation'?; 12. Square root error and random walks; 13. What is Riemann's hypothesis?; 14. The mystery moves to the error term; 15. Cesaro smoothing; 16. A view of Li(X) - Ï (X); 17. The prime number theorem; 18. The staircase of primes; 19. Tinkering with the staircase of primes; 20. Computer music files and prime numbers; 21. The word 'spectrum'; 22. Spectra and trigonometric sums; 23. The spectrum and the staircase of primes; 24. To our readers of part I; 25. Slopes and graphs that have no slopes; 26. Distributions; 27. Fourier transforms: second visit; 28. Fourier transform of delta; 29. Trigonometric series; 30. A sneak preview; 31. On losing no information; 32. Going from the primes to the Riemann spectrum; 33. How many θi's are there?; 34. Further questions about the Riemann spectrum; 35. Going from the Riemann spectrum to the primes; 36. Building Ï (X) knowing the spectrum; 37. As Riemann envisioned it; 38. Companions to the zeta function.

About the Author

Barry Mazur is Gerhard Gade University Professor of Mathematics at Harvard University, Massachusetts. He is the author of Imagining Numbers: (Particularly the Square Root of Minus Fifteen) and co-editor, with Apostolos Doxiadis, of Circles Disturbed: The Interplay of Mathematics and Narrative. William Stein is Professor of Mathematics at the University of Washington. Author of Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach, he is also the founder of the Sage mathematical software project.


'This is an extraordinary book, really one of a kind. Written by two supreme experts, but aimed at the level of an undergraduate or curious amateur, it emphasizes the really powerful ideas, with the bare minimum of math notation and the maximum number of elegant and suggestive visuals. The authors explain why this legendary problem is so beautiful, why it is difficult, and why you should care.' Will Hearst, Hearst Corporation 'This book is a soaring ride, starting from the simplest ideas and ending with one of the deepest unsolved problems of mathematics. Unlike in many popular math books puffed up with anecdotal material, the authors here treat the reader as seriously interested in prime numbers and build up the real math in four stages with compelling graphical demonstrations revealing in deeper and deeper ways the hidden music of the primes. If you have ever wondered why so many mathematicians are obsessed with primes, here's the real deal.' David Mumford, Brown University, Rhode Island 'This is a delightful little book, not quite like anything else that I am aware of ... a splendid piece of work, informative and valuable. Undergraduate mathematics majors, and the faculty who teach them, should derive considerable benefit from looking at it.' Mark Hunacek, MAA Reviews 'This book is divided into four parts, and succeeds beautifully in giving both an overview for the general audience and a sense of the details needed to understand how quickly the number of primes grows. This is accomplished through a very clear exposition and numerous illuminating pictures.' Steven Joel Miller, MathSciNet 'Where popularizers of mathematics usually succumb either to a journalist's penchant for 'man bites dog' irony and spectacle or a schoolteacher's iron will to simplify away the terror, one might call the distinctive approach here 'take a lay reader to work'. Computers now provide mathematicians a laboratory, and the authors exploit this modern power to exhibit graphics, making the key equivalence a luminous phenomenon of experimental mathematics ... for its clarity and the importance of its topic, this book deserves the same classic status as A Brief History of Time (CH, Jul'88). Summing Up: Essential. All readers.' D. V. Feldman, CHOICE 'Prime Numbers and the Riemann Hypothesis is an agile, unusual book written over a decade, one week per year; it can be considered a sort of collaborative work, in that each version was put online with the purpose of getting feedback.' Massimo Nespolo, Acta Crystallographica Section A: Foundations and Advances '... a great gift for a curious student. Using the graphical methods found in calculus reform texts, this beautiful little book allows a patient reader with a good grasp of first-year calculus to explore the most famous unsolved problem in mathematics, the so-called Riemann Hypothesis, and to understand why it points to as yet undiscovered regularities in the distribution of prime numbers.' Donal O'Shea, The Herald Tribune

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