European Mathematical Society - 11M26
https://euro-math-soc.eu/msc-full/11m26
enThe Riemann Hypothesis. A Million Dollar Problem
https://euro-math-soc.eu/review/riemann-hypothesis-million-dollar-problem
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a translation of the Dutch book <em>De Riemann-hypothese: Een miljoenenprobleem</em> 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.</p>
<p>
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.</p>
<p>
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.</p>
<p>
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.</p>
<p>
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 $\mathrm{Re}(z)=1/2$.</p>
<p>
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 $0< |z|< 1$. </p>
<p>
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 <a href="/review/prime-numbers-and-riemann-hypothesis" target="_blank">Prime Numbers and the Riemann Hypothesis</a> 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.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is the English version of a booklet that the authors originally published in Dutch after they had given an intensive online course on this subject. They organized it four years in a row for interested secondary school students. Their course (and also this text) requires hard working since many exercises are provided that should be solved (the book gives solutions at the end) to properly assimilate the material. It brings the readers to a level of understand what the Riemann hypothesis is. They even prove the prime number theorem. All the mathematics that does not belong to the standard curriculum of that age (infinite series, complex analysis,...) is provided as far as needed for their purpose.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/roland-van-der-veen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Roland van der Veen</a></li><li class="vocabulary-links field-item odd"><a href="/author/jan-van-de-craats" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jan van de Craats</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/maa-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MAA Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780883856505 (pbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£32.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">155</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.maa.org/press/ebooks/the-riemann-hypothesis" title="Link to web page">http://www.maa.org/press/ebooks/the-riemann-hypothesis</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11m26" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11M26</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</a></li></ul></span>Sat, 04 Mar 2017 11:23:25 +0000Adhemar Bultheel47499 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/riemann-hypothesis-million-dollar-problem#commentsPrime Numbers and the Riemann Hypothesis
https://euro-math-soc.eu/review/prime-numbers-and-riemann-hypothesis
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
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.</p>
<p>
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.</p>
<p>
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.</p>
<p>
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.</p>
<p>
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.</p>
<p>
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.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This wonderfully illustrated booklet explains for the lay person in very slow pace what the Riemann hypothesis actually is, why it is important, and what kind of partial results are known so far. It starts with the most elementary concepts of what prime numbers are and builds up to the full formulation of the hypothesis. It will depend on the mathematical background of the reader how far she will be able the keep up with the mathematics.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/barry-mazur" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Barry Mazur</a></li><li class="vocabulary-links field-item odd"><a href="/author/william-stein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">William Stein</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781107499430 (pbk), 9781107101920 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£17.99 (pbk), £39.99 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">154</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107101920" title="Link to web page">http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9781107101920</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11m26" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11M26</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11a41" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A41</a></li></ul></span>Tue, 21 Jun 2016 10:48:52 +0000Adhemar Bultheel47012 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/prime-numbers-and-riemann-hypothesis#comments