Customer Reviews

The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics

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One of the attractions of number theory is that it has to do with the counting numbers; if you can get from one to two and then to three, you are well on your way to hitting all the subject matter of "The Queen of Mathematics." All those numbers can be grouped into two simple categories. The composite numbers, like 15, are formed by multiplying other numbers together, like 3 and 5. The prime numbers are the ones like 17 that cannot be formed by multiplying, except by themselves and 1. Those prime numbers have held a particular fascination for mathematicians; they are the atoms from which the composites are made, but they have basic characteristics that no one yet has fully fathomed. We know a lot about prime numbers, because mathematicians have puzzled over them for centuries. We know that as you count higher and higher, the number of primes thin out, but Euclid had a beautiful proof that there is no largest prime. However, the primes seem to show up irregularly, without pattern. Can we tell how many primes are present below 1,000,000 for instance, without counting every one? How about even higher limits? Speculating about the flow of primes led eventually to the Riemann Hypothesis, the subject of _The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics_ (HarperCollins) by mathematician Marcus du Sautoy. The counting numbers turn out to be astonishingly complicated, and Du Sautoy knows that egghead number theorists will understand these complications better than we nonmathematicians, but he invites us to consider at a layman's level the importance of the particular quest of proving the Riemann Hypothesis. He is convincing in his demonstration that it is worth knowing what all the effort is about.

Bernhard Riemann, a mathematician at the University of Gottingen, introduced a "zeta function," and proposed that when this particular function equals zero, all the zeros will wind up on a specific line when graphed on the complex plane. Further effort has shown that there are millions of zero points on that line, just as the hypothesis says, and no zero points have been found off the line. Neither of these facts makes a proof, however. Du Sautoy wisely shows some of the enormously complex technicalities of the speculations and computations, but makes no attempts to try to get the reader to comprehend the hypothesis at the level he does. There are a number of reasons that the proof is so important. Right now there are a large number of tentative proofs of important mathematical ideas; they are all based on the Riemann Hypothesis being true, but of course, it has not itself been proved. A proof would tell us more about the prime distribution and finding primes, and this subject has become vital since cryptography, including how you privately send your credit card number across the internet, is based on prime numbers and the difficulty of factoring two big primes multiplied together. The way the Riemann zeros are distributed seems to mirror the patterns quantum physicists find among the energy levels of the nuclei of heavy atoms; in proving Riemann, we may have a closer understanding of fundamental reality.

With the Riemann Hypothesis central to a lot of mathematical effort, Du Sautoy is able to bring in a lot of side issues, such as Turing's attempt to find a program that would attack the proof, the four color map theorem and computer proofs in general, Gödel's Incompleteness Theorem, and much more. The mathematics, such as it is, is leavened by portraits of mathematicians, who range from conventional to very peculiar. A good deal is said about the dashing Italian mathematician Enrico Bombieri who rocked the mathematical world with the announcement that the Riemann Hypothesis had finally been proved. There was jubilation over the announcement until mathematicians realized that the e-mail bore the date 1 April. He could not have picked a better theme for an April Fool's joke; all the mathematicians were eager to see this one proof finally nailed down. Readers who take du Sautoy's entertaining tour can get an idea of why all the effort is being expended on the proof, and what elation there will be if it is ever found.

This is an exceptionally interesting book on the nature of prime numbers. The author succeeds on two fronts, he makes an incredibly vexing mathematical problem understandable to the lay person, AND he successfully explains most of the attacks against the problem for the last 150 years in a way that is both intrigueing and understandable. This is NOT a book with pages and pages of formulae, but it does contain a rich description of this problem which helps make it accessible to the curious mind.

The author has provided an excellent index at the back of the book for people that want to delve further. In addition, the author mentions several websites in the book that are helpful. The book contains many interviews with people currently working in the field to solve this problem .. but what I found most interesting, was how far ahead of his time Riemann himself was. The fact that he was able to come up with this hypothesis way before the advent of modern computational equipment and the ability to compute the zeroes necessary in the formula ... truly marks him as a unique mind. What would he be like if he lived today, with our supercomputers and other aids to computation?

I felt the book was very thought provoking on several fronts, the author's style was quite accessible, and it was enjoyable reading.

As the previous reviewers have already noted, du Sautoy does a great job bringing together the history of research that has been done on prime numbers, especially the Riemann Hypothesis and anything that pertains to that problem. I had not heard of the physics connection until I read this book and I did enjoy reading about it. The coverage is also very comprehensive and very thorough.

The bad is the purple prose that du Sautoy resorts to in order to make the material accessible to the lay reader. i think perhaps he underestimates his audience -to some a fatal flaw, to others a grating annoyance. My opinion is somewhere in between. It is rather difficult to express higher mathematics in a language other than in the mathematical language. I thought he did a pretty decent job with many of the concepts but I wonder what Simon Singh could have done with the same information. For example, du Sautoy's explanation of the RSA encryption method was lightweight and confusing. I think I had to read the pages four or five times before I saw how he was trying to explain the method. I am not a mathematician but I do have extensive background in mathematics, so if I got confused, what happens to the average reader?

The ugly is the way he flits around in his narrative. There is never any sense of when he is done talking about one development and the beginning of another. the history of the mathematicians were cursory at best. I understand that the purpose is to explore the idea of primes and their frequency but I agree also that the history and quirks of the mathematicians are interesting sidenotes that help the narrative move along, but don't leave the reader hanging!!!

regardless, I would recommend the book because of the expanse of mathemtical ground covered and the interesting concept introduced. I like the concept, I just did not care for the execution.

Mathematical texts are seldom page-turners, but du Sautoy has written a book that's difficult to put down. He describes the history of the Riemann Hypothesis as an unfolding mystery, and there are genuine cliffhangers that leave you wondering what twists and turns the plot is about to take.

My favorite part of the book, though, consists of the characters. Instead of dryly listing each mathematician's achievement, du Sautoy describes their personalities and quirks. If you have a background in math, you'll have heard of most of the mathematicians in this book, but perhaps not known which were womanizers, which were rivals with each other, and which were just plain nuts. I'd wager that this is the first math text ever written to start with the description of an April Fool's prank.

This plot and character development means there's not as much space for technical explanations as one might like, and du Sautoy consistently avoids technical details to emphasize the ideas behind them instead. For me this was fine, since it's given me the motivation to read a more technical book, but folks who are interested only in the math behind the Riemann Hypothesis and nothing more would be happier selecting another text.

Neat historic and personality stuff and mathematical overview, but if you like and want to learn the math, the book is very frustrating and somewhat offensive. Du Satoy omits almost all mathematial rigor, and most mathematical detail.

He frequently uses metaphors, usually without ever telling what the real mathematical terms are. To use a metaphor of my own, it reads sort of like, "If you think of numbers as flocks of birds, and prime numbers as differently colored birds, then the Riemann Hypothesis is like a fish." Something neat is going on, but I don't know what it is.

He never actually tells what the Riemann Hypothesis is in clear terms, or even what Li(x) is in the Prime Number Theorem.

I also dislike many of the metaphors themselves. After a while I couldn't stand it any more, and skipped every paragraph containing the words "clock calculator," which he uses instead of "modular number systems." Even as metaphors go, it's horrible. Calculators are little boxy things with number keys on them, and are nothing like modular number systems. Later in the book, I skipped paragraphs which said "quantum drum." I never did figure out what he meant by this.

This could all have been so easily fixed by a short appendix with the actual equations, or even by a list of references. (The book does have references, but they're also non-mathematical, and furthermore they aren't described beyond their titles.)

He even has a website for the book, where this additional information could have gone, but it's just more of the same. It's also, by the way, of the "pretty is more important than navigable" school of website design.

An appendix to the book with a timeline and a list of the mathematicians and their accomplishments would have also been invaluable--how much of this book will you remember in six months?

He sometimes plays fast and loose with mathematical details, e.g., referring to complex numbers as imaginary numbers. It probably doesn't matter, but you have to wonder what else you can't trust.

And some of the stuff, presented without explanation, just doesn't make sense. For instance, (I'm not making this up, though I am paraphrasing): "Ramanujan sent Hardy a letter which said 1+2+3+... = -1/12. At first Hardy and Littlewood thought this was the work of an idiot, but then they realized it said 1+2+3+... = -1/12, and he was actually brilliant." (pp.135-137, paperback edition)

Every once in a while there was some unambiguous math. I liked Euler's product for changing the zeta function into a product of series of reciprocal powers of primes, and how for the harmonic series this simplifies down to a proof that there are infinitely many primes. (p.81). I also like, and am astounded by and don't understand, the equation which generates all the primes but is useless on p.200.

Another neat thing I took from the book was the evidence that individual, unique genius exists, that ideas aren't just out there waiting to be discovered by the next mathematician who comes along. Riemann knew stuff which still isn't know to this day (and he apparently wrote it down, in a black book which disappeappeared but might still exist somewhere). It was nice to have confirmation of this, since it turns up in science fiction ("if we destroy all his notes, the human race will never rediscover this"). I was always skeptical, but I guess it's true.

The main reason I gave the book four stars is that it made me interested enough to want to find another book and look up the real math.

This book is quite wonderful. It is the first non-scientific book I have read this summer and I could not put it down. Dr. du Sautoy has a wonderfully light style, which makes the search for a proof of Riemann's Hypothesis as exiting as the best mystery story. What a story it is extending as it does over four centuries and taking place at the many of the worlds greatest centers of learning. It you are going to read one book on mathematics this is the one to read!

This book "tells a story of eccentric and brilliant men" who have investigated the Riemann hypothesis and related matters, and tells it fairly well. If it were solely about the mathematicians, I might well have given four stars; I know very little independent information about them with which to check the author's story.

But it tries to tell the story of the mathematics as well, though perhaps half-heartedly, and does not succeed at all well. Such recurring metaphors as "Nature has picked the primes" seem rather strange to me, but there are quotations from other mathematicians that appear to state an opposing view.

It is not especially well written. It is written in British rather than American English, which is perfectly all right in itself, and possibly accounts for "protest at" where "protest" would suffice. Such analogies as "ley lines" might fly better overseas. But "explanations for natural phenomenon" doesn't work anywhere. In one sentence he writes of "irrational numbers" and "numbers with a non-repeating decimal expansion" as if they were different entities. Earlier he hazily defines irrational numbers without using the word "ratio", allowing the questionable assertion "the name reflected mathematicians' sense of unease". At one point he writes "N and N+1 cannot both be prime ... because at least one of these numbers is divisible by 2"; of course, exactly one is divisible by 2. Although the seventh graders I have worked with are familiar with such terms as "integer" and "modulus", he muddies many a description by refusing to use them and many similar widely understood terms. Although the term "factorise" (British for the verb "factor") is used, he can still write such cumbersome lines as "the fast way to crack numbers into the primes from which they are built" when he means simply "the fast way to factorise".

He describes the Ishango bone: "Marked on it are three columns of four notches. In one of the columns we find 11, 13, 17 and 19 notches...." How did four notches change? An appendix at least gives a website where the bone can be viewed. The same appendix gives the author's own website, which appears to be inoperative.

While it may be hopeless to explain all the math in the book, the author could have done more. He states Hilbert's seventh problem, and later states that it was solved, but does not say when, by whom, or even whether the answer was yes or no. On page 200 he gives a fascinating formula in 26 variables which generates all primes but no other positive number, but fails to state what values the variables can take (presumably non-negative integers). He adds, "some choices ... give a negative answer, and we just have to ignore them"; from what I can see, the googol non-negative choices for these variables with lowest sum will generate painfully few primes, so his comment is startlingly understated. Nonetheless, it was nice to see such a formula in print; it hadn't even been discovered when I got a degree in math many years ago. Possibly non-mathematicians will find this book's shortcomings less annoying than I have, and as a retired mathematician I certainly enjoyed such lines as "a trance is in fact very close to the state of mind that most mathematicians try to achieve".

+++++

This twelve chapter book by Oxford mathematics professor, Marcus du Sautoy, introduces the reader to the fascinating universe of prime numbers. At the same time, we enter the world of the mathematician, a world unknown to most readers.

What are prime numbers? A prime is simply a number that cannot be divided by any other number except by one and itself. For example, the primes up to fifty are as follows:

{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47}.

This sequence of fifteen prime numbers and even those primes beyond fifty appear to be random. If you imagine the position of each prime being a note in a musical song, then you get, as this book's title states, "the music of the primes." But for centuries this music was "disorganized noise" since, as mentioned, the prime numbers appeared to be random leaving such questions as these:

(1) How can one predict when the next prime will occur?
(2) Is there a formula that can generate prime numbers?
(3) Is there a pattern to the apparent randomness of primes?

By the mid-1800s, a mathematician named Bernhard Riemann (1826-1866), "the Wagner of the mathematical world," came as close as anyone to solving this problem of prime randomness. He presented an educated guess (known as a hypothesis) that the primes may not be really that random and that there actually may be a "harmony" between them and other numbers. In other words the music of the primes may not be as disorganized as once thought. Indeed, "nature [may have] hidden in the primes the music of some mathematical orchestra."

Trouble is that Riemann never proved his hypothesis and ever since, this proof, as this book's subtitle states, has become "the greatest mystery in mathematics." This book is all about the search for that proof. Du Sautoy presents the "prime suspects" (pun intended) in this search to discover the proof of the Riemann Hypothesis. (Note that anyone in today's world that can discover the proof will win a million dollars.)

Du Sautoy describes this quest to find a proof more eloquently: "We [will crisscross] the historical and physical world: Napoleans's Revolutionary France; the neo-humanistic revolution of Germany, from grand Berlin to the medieval streets of [the small town of] Gottingen; the strange alliance between Cambridge [University in England] and India; the isolation of war-torn Norway; the New World, and a new academy founded in Princeton [University] for those brave seekers [seeking the proof] of Riemann's [Hypothesis] expelled from Europe by the ravages of war; and finally to modern Paris and a new [mathematical] language, first [discovered] in a prison cell and which [caused psychological unrest in] the mind of one of its key developers."

In order to keep this quest interesting, du Sautoy presents interesting stories and phenomena along the way. For example, the author describes two species of cicada (large insects) that have prime number life cycles of 13 and 17 years. He goes on to explain why each species chose a prime number of years as the length of their life cycle.

"The story of the primes spreads well beyond the mathematical world" from quantum physics to computer security. With respect to computer security, "The primes now affect all our lives as they protect the world's electronic secrets from the prying eyes of Internet hackers."

There are two things I appreciated about this book. First, the mathematics. You don't have to be a math whiz to read this book (even though I found it helpful to slow down and read the math sections carefully). Important mathematical concepts are explained and complex mathematics is made understandable (which is no small feat). All this is aided by graphs and tables. Enough math is presented (actually more math than I expected is presented) so that the reader has a clear understanding of relevant concepts. (Some people think that not enough math is presented. Personally, I don't understand this. As the book's subtitle states, this book is about a search and not solely about complex mathematics. If I wanted a book on complex mathematics, I would have bought one.) Second, du Sautoy's enthusiasm comes through as he details the search for a proof. For me, this enthusiasm was infectious and I found myself caught up in the story.

Finally, this book has more than twenty-five black-and-white photographs and more than fifteen helpful graphs and tables. There is a "Further Reading" section "for those who have been stimulated to dig deeper into [this] subject." Also, there is a list of informative Internet sites.

In conclusion, this is a book for those who not only have a prime obsession but also for those who have a general interest in mathematics and its mysteries. This is the prime book to read if you want to understand the music of prime numbers and to follow the mathematical search "to solve the greatest mystery in mathematics." Also, you can read this book anytime not just in prime time!

+++++

Warning!

This book could change your life.

A well written history of the relationship between the strange, seemingly random series of prime numbers. An essential read for anyone interested in the why and wherefore of the construction of a naturally occuring number series.

This book gives a good historical account of the Riemann Hypothesis. The book is not too technical and is filled with historical notes which are in themselves fascinating. I also found the relation of quantum physics (energy level distribution of certain nuclei) to the Riemann Hypothesis (distribution of the Zeta function zeroes along the imaginary axis w/Re(1/2)) very interesting. The mention of elliptic curves as the basis of a data encryption algorithm was also of interest. In short, this book has succeeded in peaking my curiosity on a variety of mathematical subjects. I'm only giving it 4 stars since I would have liked a more technical treatment of the subject.