# Prime Numbers and the Riemann Hypothesis

The Riemann hypothesis is currently, now that Fermat's Last Theorem has been proved, the unsolved problem in mathematics that has been researched most, both theoretically and experimentally. Many known `theorems' start with `If the Riemann hypothesis holds then...' and often it also holds the other way around. Thus there are many equivalent formulations. Contrary to FLT, its formulation is not that easy to understand for the non-mathematician. The authors have chosen to give several formulations of the hypothesis, starting from the most direct elementary form: how many prime numbers are there less than a certain number, and gradually formulating equivalent forms that are more and more mathematical: from the staircaise function $\pi(x)$, counting prime numbers up to $x$, until the well known formulation about the location of the zeros of the zeta function.

The authors have written the book in four parts. The first one takes about half of the booklet and goes through the historical development from elementary prime number concepts to Riemann's use of Fourier analysis to understand the spectrum of the prime number distribution. This is intended for any interested reader. Mathematics are not or only present in disguised form. All the difficult technicalities and the frightening sharp edges are nicely hidden. The reader is however treated with dignity, i.e., (s)he is not considered as a complete idiot. All what is needed is some interest in learning more about the problem at hand. For example it is explained how the logarithms enters the scene, what the logarithmic integral is, and how it was used by Gauss to approximate the prime number distribution, and how in Fourier analysis one represents an arbitrary function as a summation of cosines.

Part II is described by the authors as a preparation to extensions of Fourier analysis as needed in the next parts for readers who had at least one calculus course. This is indeed needed in the next part where the link is made between the location of the prime numbers and the Riemann spectrum. The step is not trivial. Approximating a smooth function by a sum of cosines is one thing, but approximating a staircase function stepping at all the integer muptiples of prime numbers requires distributions, which is itself already a difficult concept. But the reader is convinced that the idea works, not so much by the theory but by the graphics of experiments that show the spikes appearing at the appropriate places where primes or their powers should appear. To appreciate part IV the reader is assumed to have some knowledge about complex functions because it comes to the final description of the hypothesis as the statement about the location of the nontrivial zeros of the Riemann zeta function on the $x=1/2$ axis in the complex plane. Here one needs to introduce the concept of analytic continuation for the summation of infinite power series with complex exponents, and to link this with infinite products involving primes. Especially when one has to link zeros of the zeta function to the spectrum introduced before, it may become a bit fuzzy for the reader that is not properly prepared.

Although there are already many references in the text, there is a kind of appendix with endnotes which give further references, often links to an internet site or to a pdf where the full paper that is referred to can be downloaded. These notes are also used to give extra technical explanation. At several places these are really essential, certainly in the later parts, if you do want to get to the mathematics.

The chapters are very short, sometimes just one paragraph, so that the reader is brought to the next level teaspoon by teaspoon. And yet the reader is introduced to random walks, Cesàro summation, and to Fourier analysis, but also to distributions, and how they can be used in Riemann's Fourier approach. There are also many graphics clearly showing the approximations for the staircase of primes and how these look at different scales. These are essential in the concept of the book. They strongly contribute to the understanding of what is going on. The zeta function comes surprisingly late into the picture, or maybe not so surprising since this is as far as the authors want to bring the reader.

Besides the graphs, there are many other illustrations of the main mathematicians involved, historical as well as contemporary. The booklet is published on glossy paper. It is not recommended to buy the paperback edition because the pages easily get detached from the cover, a very unfortunate property, since very soon you will end up with a set of loose pages and a separate cover instead of a nice book. Since the book is thin enough, a hard cover does not seem a good option either. However, the authors did a wonderful job. Given its compactness and the richness in content, this is a marvelous booklet. It does exactly what the authors intended to: introduce the reader to the problem. You will absolutely not find here a springboard to the mathematics needed to solve the problem. Thus you will not learn how to tackle the problem, in fact nobody currently knows how to solve it, but you will learn about the standard mainstream approach so far. So not how to proceed in the future, but a short history and an idea about the what and why of the Riemann Hypothesis is expertly explained. Whether or not you have some mathematical background, you can pick the level that suits you.

**Submitted by Adhemar Bultheel |

**21 / Jun / 2016