Book contents
- Frontmatter
- Contents
- Preface
- PART I The Riemann Hypothesis
- 1 Thoughts About Numbers
- 2 What are Prime Numbers?
- 3 “Named” Prime Numbers
- 4 Sleves
- 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 Cesàro 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
- PART II Distributions
- PART III The Riemann Spectrum of the Prime Numbers
- PART IV Back to Riemann
- Endnotes
- Index
24 - To Our Readers of Part I
from PART I - The Riemann Hypothesis
Published online by Cambridge University Press: 05 May 2016
- Frontmatter
- Contents
- Preface
- PART I The Riemann Hypothesis
- 1 Thoughts About Numbers
- 2 What are Prime Numbers?
- 3 “Named” Prime Numbers
- 4 Sleves
- 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 Cesàro 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
- PART II Distributions
- PART III The Riemann Spectrum of the Prime Numbers
- PART IV Back to Riemann
- Endnotes
- Index
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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- Chapter
- Information
- Prime Numbers and the Riemann Hypothesis , pp. 67 - 68Publisher: Cambridge University PressPrint publication year: 2016