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The Riemann Hypothesis

The Reimann Hypothesis

Roland van der Veen and Jan van de Craats

Catalog Code: NML-46
Print ISBN: 978-0-88385-650-5
Electronic ISBN: 978-0-88385-989-6
154 pp., Paperbound, 2016
List Price: $45.00
Member Price: $33.75
Series: Anneli Lax New Mathematical Library

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This book introduces interested readers to one of the most famous and difficult open problems in mathematics: the Riemann hypothesis. Finding a proof will not only make you famous, but also earns you a one million dollar prize. The book originated from an online internet course at the University of Amsterdam for mathematically talented secondary school students. Its aim was to bring them into contact with challenging university level mathematics and show them why the Riemann Hypothesis is such an important problem in mathematics. After taking this course, many participants decided to study in mathematics at university.

Table of Contents

1. Prime Numbers
1.1 Primes as elementary building blocks
1.2 Counting Primes
1.3 Using the logarithm to count powers
1.4 Approximations for \(\pi(x)\)
1.5 The prime number theorem
1.6 Counting prime powers logarithmically
1.7 The Riemann hypothesis—a look ahead
1.8 Additional exercises
2. The zeta function
2.1 Infinite sums
2.2 Series for well-known functions
2.3 Computation of \(\zeta(2)\)
2.4 Euler’s product formula
2.5 Looking back and a glimpse of what is to come
2.6 Additional exercises
3. The Riemann hypothesis
3.1 Euler’s discovery of the product formula
3.2 Extending the domain of the zeta function
3.3 A crash course on complex numbers
3.4 Complex functions and powers
3.5 The complex zeta function
3.6 The zeroes of the zeta function
3.7 The hunt for zeta zeroes
3.8 Additional exercises
4. Primes and the Riemann hypothesis
4.1 Riemann’s functional equation
4.2 The zeroes of the zeta function
4.3 The explicit formula for \(\psi(x)\)
4.4 Pairing up the non-trivial zeroes
4.5 The prime number theorem
4.6 A proof of the prime number theorem
4.7 The music of the primes
4.8 Looking back
4.9 Additional exercises
Appendix A. Why big primes are useful
Appendix B. Computer support
Appendix C. Further reading and internet surfing
Appendix D. Solutions to the exercises
Index

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