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

Features

• Cover Type: Paperback with 352 pages
• Published by: Harper Perennial April 27, 2004
• Written in: English
• ISBN 10 Number: 0060935588
• ISBN 13 Number: 978-0060935580
• Book Dimensions: 10.7 x 8.1 x 0.8 inches
• Weighs: 10.6 ounces

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. forty illustrations, charts and photos.
Copyright 2003 Reed Business Information, Inc. --This text refers to an out of print or unavailable edition of this title.

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 greater 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). --This text refers to an out of print or unavailable edition of this title.

Reader Reviews
This review is from: The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics (Hardcover) 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.