# The Riemann Hypothesis. A Million Dollar Problem

This is a translation of the Dutch book *De Riemann-hypothese: Een miljoenenprobleem* published by Epsilon Uitgaven in 2011. It grew out of an intensive online course given by the authors in the period 2006-2010 for talented secondary school students. The course lasted four weeks and students got support for solving the many exercises via the internet.

The idea of the course is fully maintained in this text. It has four parts, corresponding to the four weeks of the course. Its material is accessible for interested secondary school students and, why not, for starting university students as well. It is however not a leisurely reading text. Serious working and solving the many exercises that are sprinkled throughout the text is required. No more online support for the book but solutions are provided in an appendix at the end. For readers who are in for a challenge, there are somewhat more demanding exercises at the end of each part. Three other short appendices refer to external sources. One briefly gives a sketch of why large prime numbers are important. Basically it just mentioning that they are used in the RSA cryptosystem. Another one refers to freely available software packages. Besides the commercial algebra packages like Maple and Mathematica, the free software proposed can be found on the web page of Wolfram Alpha. The necessary commands to solve some of the exercises are given too. For more intensive computations, one is referred to the Sage website. A third appendix lists four books and a number of websites for further reading and experimenting.

The contents of the four parts does not bring big surprises. The first one of course has to start with prime numbers and introduces the prime counting function $\pi(x)$, counting the number of primes less than $x$. Some experiments to approximate this staircase soon leads to the idea that logarithms must be involved. The prime number theorem $\pi(x)\sim x/\log(x)$ soon pops up, but a sketch of the proof has to wait till the end of the book. Another choice is Chebyshev's function $\psi(x)=\sum_{p\le x}\lfloor\log_p x\rfloor \log p$ where the sum is over the primes $p$. Because in this one, the primes are weighted depending on the number of their powers less than $x$, this $\psi(x)$ is almost a straight line. This way of weighting the primes when counting them is of course is an essential element in the analysis of Riemann. So a sneak preview of the hypothesis is the end of the first part.

In the second part, the key player is the Riemann zeta function $\zeta(x)=\sum_{k\in\mathbb{N}} k^{-x}$. In order to introduce this properly, a discussion is needed to define infinite sums and functions defined by power series. In order to evaluate $\zeta(2)=\pi^2/6$, an infinite product for the sinc function is derived. The cliffhanger for this part is Euler's product formula that links the zeta function to the primes.

A sketch of the proof of this Euler formula $\zeta(x)=\prod_{p~\mathrm{prime}} 1/(1-p^{-x})$ is the start of part three. The zeta function is however taking over again since it needs to be extended to the whole complex plane. This requires a crash course on complex numbers and complex functions. An elementary form of analytic continuation allows to define $\zeta(z)$ for all complex $z\ne1$. The end of this part is again a forward reference to the next one announcing the trivial and nontrivial zeros of $\zeta(z)$. With these defined, it finally becomes possible to fully understand the meaning of the formulation of the Riemann hypothesis: all the nontrivial zeros of the zeta function are on the critical line $\mathrm{Re}(z)=1/2$.

In part four all the efforts come to a conclusion. The $\psi(x)$ function can be expressed as $x-\ln(2\pi)$ plus some correction. And using Euler's formula, the correction can be expressed as a sum over the zeros of the function $\zeta(z)$. The trivial zeros $-2k$ are easily obtained via Riemann's functional equation and the part in $\psi(x)$ corresponding to these trivial zeros can be summed up to give $−\frac{1}{2}\ln(1−x^{−2})$. So the remaining sum is related to the nontrivial zeros, which is the core issue of the Riemann hypothesis. The book culminates in a proof of the prime number theorem along the lines of the proofs by Hadamard and de la Vallée Poussin by showing that all the nontrivial zeros are strictly inside in the critical strip $0< |z|< 1$.

It is clear that the text is quite a challenge for secondary school students, but with some elementary introductions to topics that do not belong to their standard curriculum, they are brought a long way on the road to understand the Riemann hypothesis. Although infinite cosine series do appear in the text, the text stops on the verge of where Fourier analysis needs to take over. At least, Fourier analysis is not formally introduced. That is where Mazur and Stein in their version of Prime Numbers and the Riemann Hypothesis push the limit a bit further. It is a marvelous idea to bring young students this far on the scale of mathematics. What I am a bit missing is the importance of the Riemann hypothesis. Explaining the RSA encryption with some details would of course requiring another booklet of this type, but just mentioning it briefly is not really bringing the insight or making the importance of proving the hypothesis very concrete. But of course one has to draw the line somewhere, and there are other popular books around where one can read more about RSA and other wonderful things about prime numbers. This booklet is a wonderful guide when teachers around the globe want to stimulate the interest in mathematics or explain what pure mathematicians in the 21st century are working on. Perhaps they might think of starting up a similar course as the authors of this book did. The latter claim that several of the students that attended their course afterwards decided to start a mathematics education at the university.

**Submitted by Adhemar Bultheel |

**4 / Mar / 2017