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

~ (Author) "One hot and humid morning in August 1900, David Hilbert of the Uni of Gottingen addressed the International Congress of Mathema in a packed lecture..." (more)
Key Phrases: zeta landscape, zero off the line, factorising numbers, Fermat's Last Theorem, Prime Number Theorem, Riemann's Hypothesis (more...)

Editorial Reviews

From Publishers Weekly

The quest to bring advanced math to the masses continues with this engaging but quixotic treatise. The mystery in question is the Riemann Hypothesis, named for the hypochondriac German mathematician Bernard Reimann (1826-66), which ties together imaginary numbers, sine waves and prime numbers in a way that the world's greatest mathematicians have spent 144 years trying to prove. Oxford mathematician and BBC commentator du Sautoy does his best to explain the problem, but stumbles over the fact that the Riemann Hypothesis and its corollaries are just too hard for non-tenured readers to understand. He falls back on the staples of math popularizations by shifting the discussion to easier math concepts, offering thumbnail sketches of other mathematicians and their discoveries, and occasionally overdramatizing the sedentary lives of academics (one is said to be a "benign Robespierre" whose non-commutative geometry "has instilled terror" in his colleagues). But du Sautoy makes the most of these genre conventions. He is a fluent expositor of more tractable mathematics, and his portraits of math notables-like the slipper-shod, self-taught Indian Srinivasa Ramanujan, a mathematical Mozart who languished in chilly Cambridge-are quite vivid. His discussion of the Riemann Hypothesis itself, though, can lapse into metaphors ("By combining all these waves, Riemann had an orchestra that played the music of the primes") that are long on sublime atmospherics but short on meaningful explanation. The consequences of the hypothesis-a possible linkage to "quantum chaos," implications for internet data encryption-may seem less than earth-shaking to the lay reader, but for mathematicians, the Riemann Hypothesis may be the "deepest and most fundamental problem" going. 40 illustrations, charts and photos.
Copyright 2003 Reed Business Information, Inc.

From Scientific American

The unpredictable drip from a leaky faucet can drive almost anyone mad. Prime numbers, those divisible only by one and themselves, present a numerical equivalent. For centuries, mathematicians have tried to find a simple formula to describe where these numbers fall along the number line. But their spacing--1, 2, 3, drip, 5, drip, 7, drip, drip, drip, 11, drip, and so forth--seems to defy prediction. In 1859 German mathematician Bernhard Riemann uncovered an apparent key to unlocking the pattern, but he couldn't verify it. Many great minds have become obsessed with proving his guess, referred to as the Riemann Hypothesis (RH), ever since. Three books published in April chronicle this quest. The books cover much of the same ground, but each has a different strength. The text with the simplest title, The Riemann Hypothesis, by science writer Karl Sabbagh, provides ample hand-holding for anyone who pales at the sight of symbols or can't quite distinguish an asymptote from a hole in the graph. In Prime Obsession, by John Derbyshire, a mathematically trained banker and novelist, Riemann and his colleagues come to life as real characters and not just adjectives for conjectures and theorems. And in The Music of the Primes, written by University of Oxford mathematics professor Marcus du Sautoy, the meaning of Riemann's work unfolds by way of rich musical analogies. Why three books on the same difficult subject now? One obvious answer is that the notoriety of the RH only recently spread to circles beyond math-faculty common rooms. In 2000 the Clay Mathematics Institute (CMI), a private research organization funded by Boston banker cum math fan Landon T. Clay, offered a $1-million prize for the solution. The move won Riemann almost as many posthumous headlines as Fermat. CMI offers the one-buck bounty on seven outstanding mathematical mysteries. These so-called millennium problems are a 21st-century follow-up to German mathematician David Hilbert's famous stumpers, presented in 1900 to the Second International Congress of Mathematicians in Paris. The Riemann Hypothesis is the only problem to make both lists, a century apart--and with good reason: it is exceedingly complex, and a mounting number of results require that it be true. Timing, too, has played a part. At the end of the 18th century Carl Friedrich Gauss, one of Riemann's mentors, produced what was then the best approximation for the number of primes less than some number N--namely, N/log N. This value is sometimes too big and sometimes too small, but Gauss predicted that the error would shrink for larger Ns. By the end of the 19th century Jacques Hadamard and Charles de la Vallée Poussin proved this suggestion, called the prime number theorem (PNT). The RH was the next obvious mark. Riemann's original wording does not mention prime numbers at all but instead addresses the so-called zeta function, ?(s) = 1 + 1/2s + 1/3s + 1/4s + ... 1/ns. For s = 1, this function is the familiar harmonic series. For inputs greater than one, however, zeta becomes more exotic. Swiss mathematician Leonhard Euler discovered in the 1700s that for s = 2, zeta converges on the square of pi divided by six. It was a startling find. The decimal expansion of pi is unpredictable, and yet by way of the zeta function, it could be summed from an infinite series of neat fractions. Euler's break was the first such "zeta bridge" between seeming randomness and order. Riemann forged the next by feeding the zeta function complex numbers, those of the form a + bi, having both real and imaginary parts. These numbers were a new invention at the time. Riemann had learned about them in Paris and brought them back to Göttingen, where he studied under Lejeune Dirichlet, Gauss's successor. The older man was well acquainted with the zeta function, which he had invoked to prove one of Fermat's prime-number assertions. For Riemann, then, it was a small leap to try the new numbers in the old function. To sum up what these books take 300-plus pages to explain, Riemann homed in on points for which the zeta function fed with imaginary numbers equaled zero and viewed these "zeros" as waves--much as Euler had produced sine waves corresponding to musical notes from plugging imaginary numbers into the exponential function 100 years before. Riemann further made a connection between these waves and his own refinement of Gauss's PNT, dubbed R(N): by adding R(N) to the height of each wave above N, he could generate the exact number of primes less than N. The location of the zeros, therefore, led to that of the primes, and Riemann asserted that the zeros followed a simple pattern. They all had a real part of 1/2. In other words, were you to graph zeta, the zeros would fall along a single line. Each of the books satisfactorily presents Riemann's math--as much as it is possible to do so for a general audience--but they offer very different reading experiences. The Music of the Primes made me feel as if I were sitting through a gracefully worded lecture. The Riemann Hypothesis is more journalistic, relying on quotes from working mathematicians to tell the story. Parts of Prime Obsession read almost like a novel, others like a mathematical text. Its author, Derbyshire, segmented the book so that most of the math falls into odd chapters and the history and biographical material in even ones, but the math is as interesting as the rest. When will the RH be solved? None of the books dares to predict. Hilbert, one of the greatest mathematicians of all time, forecasted that it would happen within his lifetime. He died in 1943. In other words, it's still anyone's guess.

Kristin Leutwyler turned from the study of mathematics to journalism, serving until recently as editor of Scientific American's Web site. Now a freelance writer, she is the author of the forthcoming book The Moons of Jupiter (W. W. Norton, 2003).

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Product Details

• Hardcover: 352 pages
• Publisher: HarperCollins; 1ST edition (April 29, 2003)
• Language: English
• ISBN-10: 0066210704
• ISBN-13: 978-0066210704
• Product Dimensions: 9.1 x 5.9 x 1.3 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.5 out of 5 stars  See all reviews (27 customer reviews)
• Amazon.com Sales Rank: #447,771 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

23 of 24 people found the following review helpful:

5.0 out of 5 stars Prime Fascination, July 27, 2003

By	R. Hardy "Rob Hardy" (Columbus, Mississippi USA) - See all my reviews (TOP 50 REVIEWER)    (REAL NAME)

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.

14 of 14 people found the following review helpful:

5.0 out of 5 stars du Sautoy is Prime Time Player, June 15, 2003

By	NotToto (baltimore, MD, USA) - See all my reviews

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.

11 of 11 people found the following review helpful:

3.0 out of 5 stars The good, the bad, and the ugly, June 8, 2005

By	P. Wung "Engineering is my vocation, volleyba... (Tipp City, OH USA) - See all my reviews (REAL NAME)

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.