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.
Author(s): Roland van der Veen, Jan van de Craats
Series: Anneli Lax New Mathematical Library
Publisher: Mathematical Association of America
Year: 2017
Language: English
Pages: 157
Contents......Page 8
Preface......Page 10
1.1 Primes as elementary building blocks......Page 14
1.2 Counting primes......Page 16
1.3 Using the logarithm to count powers......Page 20
1.4 Approximations for π(x)......Page 22
1.6 Counting prime powers logarithmically......Page 24
1.7 The Riemann hypothesis—a look ahead......Page 27
1.8 Additional exercises......Page 29
2.1 Infinite sums......Page 34
2.2 Series for well-known functions......Page 39
2.3 Computation of ζ(2)......Page 42
2.4 Euler’s product formula......Page 45
2.6 Additional exercises......Page 47
3.1 Euler’s discovery of the product formula......Page 54
3.2 Extending the domain of the zeta function......Page 56
3.3 A crash course on complex numbers......Page 58
3.4 Complex functions and powers......Page 60
3.5 The complex zeta function......Page 63
3.6 The zeroes of the zeta function......Page 64
3.7 The hunt for zeta zeroes......Page 67
3.8 Additional exercises......Page 68
4 Primes and the Riemann hypothesis......Page 72
4.1 Riemann’s functional equation......Page 73
4.2 The zeroes of the zeta function......Page 76
4.3 The explicit formula for ψ(x)......Page 79
4.4 Pairing up the non-trivial zeroes......Page 82
4.5 The prime number theorem......Page 85
4.6 A proof of the prime number theorem......Page 86
4.7 The music of the primes......Page 89
4.8 Looking back......Page 91
4.9 Additional exercises......Page 94
Appendix A. Why big primes are useful......Page 100
Appendix B. Computer support......Page 104
Appendix C. Further reading and internet surfing......Page 112
Appendix D. Solutions to the exercises......Page 114
Index......Page 156
