**Anneli Lax New Mathematical Library**

Volume: 46;
2015;
144 pp;
Softcover

**Print ISBN: 978-0-88385-650-5
Product Code: NML/46**

List Price: $45.00

AMS Member Price: $33.75

MAA Member Price: $33.75

**Electronic ISBN: 978-0-88385-989-6
Product Code: NML/46.E**

List Price: $45.00

AMS Member Price: $33.75

MAA Member Price: $33.75

# The Riemann Hypothesis

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*Roland van der Veen; Jan van de Craats*

MAA Press: An Imprint of the American Mathematical Society

Recipient of the Mathematical Association of America's
Beckenbach Book Prize in 2018!

The Riemann hypothesis concerns the prime
numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
… Ubiquitous and fundamental in mathematics as they are, it is
important and interesting to know as much as possible about these
numbers. Simple questions would be: how are the prime numbers
distributed among the positive integers? What is the number of prime
numbers of 100 digits? Of 1,000 digits? These questions were the
starting point of a groundbreaking paper by Bernhard Riemann written
in 1859. As an aside in his article, Riemann formulated his now famous
hypothesis that so far no one has come close to proving: All
nontrivial zeroes of the zeta function lie on the critical
line. Hidden behind this at first mysterious phrase lies a whole
mathematical universe of prime numbers, infinite sequences, infinite
products, and complex functions.

The present book is a first
exploration of this fascinating, unknown world. It originated from an
online course for mathematically talented secondary school students
organized by the authors of this book at the University of
Amsterdam. Its aim was to bring the students into contact with
challenging university level mathematics and show them what the
Riemann Hypothesis is all about and why it is such an important
problem in mathematics.

# Table of Contents

## The Riemann Hypothesis

- Cover cov11
- Half title i2
- Copyright ii3
- Title iii4
- Epigraph iv5
- Series v6
- Contents vii8
- Preface ix10
- 1 Prime numbers 114
- 1.1 Primes as elementary building blocks 114
- 1.2 Counting primes 316
- 1.3 Using the logarithm to count powers 720
- 1.4 Approximations for π(x) 922
- 1.5 The prime number theorem 1124
- 1.6 Counting prime powers logarithmically 1124
- 1.7 The Riemann hypothesis-a look ahead 1427
- 1.8 Additional exercises 1629

- 2 The zeta function 2134
- 3 The Riemann hypothesis 4154
- 3.1 Euler's discovery of the product formula 4154
- 3.2 Extending the domain of the zeta function 4356
- 3.3 A crash course on complex numbers 4558
- 3.4 Complex functions and powers 4760
- 3.5 The complex zeta function 5063
- 3.6 The zeroes of the zeta function 5164
- 3.7 The hunt for zeta zeroes 5467
- 3.8 Additional exercises 5568

- 4 Primes and the Riemann hypothesis 5972
- 4.1 Riemann's functional equation 6073
- 4.2 The zeroes of the zeta function 6376
- 4.3 The explicit formula for ψ(x) 6679
- 4.4 Pairing up the non-trivial zeroes 6982
- 4.5 The prime number theorem 7285
- 4.6 A proof of the prime number theorem 7386
- 4.7 The music of the primes 7689
- 4.8 Looking back 7891
- 4.9 Additional exercises 8194

- Appendix A. Why big primes are useful 87100
- Appendix B. Computer support 91104
- Appendix C. Further reading and internet surfing 99112
- Appendix D. Solutions to the exercises 101114
- Index 143156