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Prime Numbers and the Riemann Hypothesis

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  • Date Published: April 2016
  • availability: Available
  • format: Paperback
  • isbn: 9781107499430

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About the Authors
  • 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 to be 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 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.

    • Provides a concise but comprehensive narrative for students and scholars alike
    • The text is broken into parts, thus providing greater accessibility
    • Offers many mathematical illustrations
    Read more


    • Honourable Mention, 2017 PROSE Award for Mathematics

    Reviews & endorsements

    "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

    'The book under review succeeds handsomely in making the case for the Riemann Hypothesis to a wide audience … Beginning with the definition of prime numbers, the authors weave their way through concrete and picturesque presentations of elementary techniques and descriptions of unsolved problems connected with the primes. They provide many insightful footnotes, concrete and illuminating figures, pointers to arXiv pages for added information, and a rich set of endnotes that contain further descriptions and details with varying levels of sophistication. After 23 short sections (a few pages each) they have arrived at a formulation of the Riemann Hypothesis in terms of counting primes up to a given size. By this point in their masterful and compelling presentation, the Hypothesis appears to be completely natural and inevitable … I have no doubt that many newcomers to the subject who have read to the end of the book will be eager to learn more and will be drawn into this fertile playground.' Peter Sarnak, Bulletin of the AMS

    'I really recommend this book if you want to get a feeling for the Riemann hypothesis without sinking into technicalities.' John Baez, The n-Category Café (

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    Product details

    • Date Published: April 2016
    • format: Paperback
    • isbn: 9781107499430
    • length: 150 pages
    • dimensions: 229 x 153 x 8 mm
    • weight: 0.26kg
    • contains: 110 b/w illus. 132 colour illus.
    • availability: Available
  • 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. Césaro smoothing
    16. A view of Li(X) - π(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 π(X) knowing the spectrum
    37. As Riemann envisioned it
    38. Companions to the zeta function.

  • Resources for

    Prime Numbers and the Riemann Hypothesis

    Barry Mazur, William Stein

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  • Authors

    Barry Mazur, Harvard University, Massachusetts
    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, University of Washington
    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.


    • Honourable Mention, 2017 PROSE Award for Mathematics
    • Winner, 2017 Choice Outstanding Academic Title

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