Veen R. van de, Craats J. van de. The Riemann Hypothesis: a million dollar problem

Washington: Mathematical Association of America, 2015. — 156 p. — (Anneli Lax New Mathematical Library, 46). — ISBN 978-0-88385-650-5, 978-0-88385-989-6.Mathematics is full of unsolved problems and other mysteries, but none more important and intriguing than the Riemann hypothesis. Baffling the greatest minds for more than a hundred and fifty years, the Riemann hypothesis is at the very core of mathematics. A proof of it would mean an enormous advance. In addition, the Riemann hypothesis was chosen as one of the seven Millennium Problems by the Clay Mathematics Institute. This means that proving the Riemann hypothesis will not only make you world famous, but also earns you a one million dollar prize.
The Riemann Hypothesis: All nontrivial zeroes of the zeta function lie on the critical line.
Hidden behind this at first mysterious statement, 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 world.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