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The Riemann Hypothesis

Roland van der Veen and Jan van de Craats
MAA Press
Publication Date: 
Number of Pages: 
Anneli Lax New Mathematical Library
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mehdi Hassani
, on

The Riemann Hypothesis is one of the hardest and most famous problems in mathematics. Its original formulation, which comes from the theory of complex functions, asserts that all non-real zeros of the Riemann zeta function have real part equal to one-half. Because of its technical formulation, it is not easy to talk about the Riemann Hypothesis without assuming knowledge of the complex function theory, but we can exploit its connections to other branches of mathematics. One of the most important is the light it sheds on the distribution of prime numbers. And there are also some elementary conjectures that turn out to be equivalent to the Riemann Hypothesis.

The book under review, which seems to be one of the first books at this level about the Riemann Hypothesis, is aimed at high school and undergraduate students. It focuses mainly on the so-called “explicit formula,” which connects the distribution of the zeros of the Riemann zeta function with the distribution of the prime numbers. The authors set the stage by explaining basic facts about complex numbers and functions, introducing the zeta function and its product formula. They then conduct numerical experiments on the explicit formula, reporting them in several figures.

The book consists of four chapters and four appendices. Each chapter ends with some exercises. The authors provide computer programming code for exercises with computational flavor and full solutions for the rest of them.

The book will be useful for students and teachers to become familiar with the Riemann Hypothesis. It may also be used as a text in a mini-course. The interested reader will not, however, find here all that could be said about the Riemann Hypothesis at this level. I believe that there would have been room in the present book for some related topics, including the many elementary statements known to be equivalent of the Riemann hypothesis, such as an inequality involving the sum of divisor function and harmonic numbers.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.

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
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