Van der Veen R., van de Craats J.The Riemann Hypothesis

Washington: Mathematical Association of America, 2017. — 157 p.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.Primes as elementary building blocks
Counting primes
Using the logarithm to count powers
Approximations for π(x)
Counting prime powers logarithmically
The Riemann hypothesis—a look ahead
Additional exercises
Infinite sums
Series for well-known functions
Computation of ζ(2)
Euler’s product formula
Additional exercises
Euler’s discovery of the product formula
Extending the domain of the zeta function
A crash course on complex numbers
Complex functions and powers
The complex zeta function
The zeroes of the zeta function
The hunt for zeta zeroes
Additional exercises
Primes and the Riemann hypothesis
Riemann’s functional equation
The zeroes of the zeta function
The explicit formula for ψ(x)
Pairing up the non-trivial zeroes
The prime number theorem
A proof of the prime number theorem
The music of the primes
Looking back
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