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24 - To Our Readers of Part I

from PART I - The Riemann Hypothesis

Published online by Cambridge University Press:  05 May 2016

Barry Mazur
Affiliation:
Harvard University, Massachusetts
William Stein
Affiliation:
University of Washington
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Summary

The statement of the Riemann Hypothesis – admittedly as elusive as before – has, at least, been expressed elegantly and more simply, given our new staircase that approximates (conjecturally with essential square root accuracy) a 45 degree straight line.

We have offered two equivalent formulations of the Riemann Hypothesis, both having to do with the manner in which the prime numbers are situated among all whole numbers.

In doing this, we hope that we have convinced you that – in the words of Don Zagier – primes seem to obey no other law than that of chance and yet exhibit stunning regularity. This is the end of Part I of our book, and is largely the end of our main mission, to explain – in elementary terms – what is Riemann's Hypothesis?

For readers who have at some point studied differential calculus, in Part II we shall discuss Fourier analysis, a fundamental tool that will be used in Part III where we show how Riemann's hypothesis provides a key to some deeper structure of the prime numbers, and to the nature of the laws that they obey. We will – if not explain – at least hint at how the above series of questions have been answered so far, and how the Riemann Hypothesis offers a surprise for the last question in this series.

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Publisher: Cambridge University Press
Print publication year: 2016

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  • To Our Readers of Part I
  • Barry Mazur, Harvard University, Massachusetts, William Stein, University of Washington
  • Book: Prime Numbers and the Riemann Hypothesis
  • Online publication: 05 May 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781316182277.025
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  • To Our Readers of Part I
  • Barry Mazur, Harvard University, Massachusetts, William Stein, University of Washington
  • Book: Prime Numbers and the Riemann Hypothesis
  • Online publication: 05 May 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781316182277.025
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • To Our Readers of Part I
  • Barry Mazur, Harvard University, Massachusetts, William Stein, University of Washington
  • Book: Prime Numbers and the Riemann Hypothesis
  • Online publication: 05 May 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781316182277.025
Available formats
×