From the Publisher

In 1859, German mathematician Bernhard Riemann presented a paper to the Berlin Academy that would forever change the history of mathematics. The subject was the mystery of prime numbers. At the heart of the presentation was an idea that Riemann had not yet proved but one that baffles mathematicians to this day.

Solving the Riemann Hypothesis could change the way we do business, since prime numbers are the lynchpin for security in banking and e-commerce. It would also have a profound impact on the cutting-edge of science, affecting quantum mechanics, chaos theory, and the future of computing. Leaders in math and science are trying to crack the elusive code, and a prize of $1 million has been offered to the winner. In this engaging book, Marcus du Sautoy reveals the extraordinary history behind the holy grail of mathematics and the ongoing quest to capture it.

The New York Times

[du Sautoy] is an insider, a research mathematician. He walks the walk and talks the talk. His discussion of mathematics is figurative and elliptical, as when mathematicians talk to one another. — James Alexander

The Los Angeles Times

Number theorist Marcus du Sautoy's book The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics compares the pattern the primes make to music. At first they seem to appear at random, like a kind of noise, but under analysis display a more musical structure. — Ben Yandell

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 Oxford-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. (Apr.) Copyright 2003 Reed Business Information.

Library Journal

Thanks to a proof by Euclid, mathematicians have known for more than 2000 years that there is no limit to the population of prime numbers; they extend to infinity. However, work continues to be done on the distribution of the primes, and much of that work now centers on efforts to prove the Riemann hypothesis. Bernhard Riemann was a great 19th-century German mathematician who offered in an 1859 paper an admittedly unproven conjecture relating some zero values of a "zeta function" to the distribution of primes. The importance of this abstruse speculation for modern research is demonstrated in a recent online search of Mathematical Reviews for the term "Riemann hypothesis"; 1403 publications were found. Now there are three more books to add to the numerous studies. Derbyshire, a mathematician by training, a member of the Mathematical Association of America, and a novelist (Seeing Calvin Coolidge in a Dream), first takes readers through well-organized mathematical fundamentals in order to give them a good understanding of Riemann's discovery and its consequences. Interspersed with the hardcore math, other chapters profile Reimann the man and trace the history of mathematics in relation to his still-unproven hypothesis. Derbyshire shows how after 150 years, the world's greatest minds still haven't found a solution. Because this book does not sugarcoat complex ideas, readers lacking at least college-level math will be hard-pressed to understand some parts. Still, this volume is highly recommended for academic and larger public libraries as an excellent introduction for nonspecialists. Du Sautoy is the only professional research mathematician among these three authors, but he does not confront his readers with very many equations or other bits of mathematical apparatus. Instead, he offers nicely done verbal descriptions of the essence of the hypothesis and the efforts to prove it. Like Derbyshire, he intersperses items from math history and from the work and interactions of current researchers. Du Sautoy's book has much to offer for most academic and public libraries, especially to readers of very limited math background. Sabbagh (A Rum Affair) has written several books on a variety of topics, not all science-related. His latest emphasizes anecdotes from contemporary mathematicians who have studied Riemann's hypothesis. Indeed, he pays so much attention to a particularly idiosyncratic mathematician, ignored by the two other authors, that in his quest for human-interest material, he seems to lose sight of serious mathematical issues. Sabbagh's discussion of the actual mathematics is not so well organized, and much of it is relegated to a series of appendixes. His book is most useful in giving readers a feel for how research mathematicians live, work, and interrelate in the 21st century. Only libraries seeking comprehensive coverage of mathematics will need to get the Sabbagh work.-Jack W. Weigel, Ann Arbor, MI Copyright 2003 Reed Business Information.

Kirkus Reviews

A Royal Society research fellow takes the Riemann Hypothesis, reputedly the most difficult of all math problems, as the focus for his lively history of number theory. Du Sautoy (Mathematics/Oxford) begins in 1900 with German mathematician David Hilbert's famous address to the International Congress of Mathematicians in Paris, where Hilbert offered 23 unsolved problems as challenges to his colleagues. Among them was the Reimann Hypothesis, which concerns the distribution of prime numbers; it is the only one still unsolved. Greek mathematicians knew that the primes are infinite in number and distributed randomly in the set of natural numbers. Two centuries ago, Carl Friedrich Gauss offered a formula to estimate how many primes lie below any given number; in 1859, Gauss's student, Bernhard Riemann, refined that estimate, based on the incredibly complex Zeta function, but died without proving his hypothesis. With a minimum of equations and mathematical symbols, du Sautoy outlines the progress each succeeding generation has made on the problem. Along the way, readers meet G.H. Hardy and J.E. Littlewood, the twin beacons of the Cambridge math department between the world wars; Ramanujan, the self-taught Indian clerk who claimed that his ideas were given to him by his family goddess; and Atle Selberg, who survived the Nazi occupation of Norway to become a leading light at Princeton's Institute for Advanced Studies. Alan Turing, the father of modern computers, tried to devise a program to attack the Riemann Hypothesis; now the primes are the key to cryptography. A Boston businessman has offered a million-dollar reward for a proof, although few mathematicians seem to need additional incentive totackle the Everest of mathematical problems. Du Sautoy keeps the story moving and gives a clear sense of the way number theory is played in his accessible text. (See Karl Sabbagh’s The Riemann Hypothesis, p. 369, which covers similar territory but spotlights current mathematicians searching for a Riemann proof.) A must for math buffs.

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Table of Contents

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The Music of the Primes

HarperCollins

Chapter One

'Do we know what the sequence of numbers is? Okay, here, we can do it in our heads .. fifty-nine, sixty-one, sixty-seven ... seventy-one ... Aren't these all prime numbers?' A little buzz of excitement circulated through the control room. Ellie's own face
momentarily revealed a flutter of something deeply felt, but this was quickly replaced by a sobriety, a fear of being carried away, an apprehension about appearing foolish, unscientific.
- Carl Sagan, Contact

One hot and humid morning in August 1900, David Hilbert of the University of Göttingen addressed the International Congress of Mathematicians in a packed lecture hall at the Sorbonne, Paris. Already recognised as one of the greatest mathematicians of the age, Hilbert had prepared a daring lecture. He was going to talk about what was unknown rather than what had already been proved. This went against all the accepted conventions, and the audience could hear the nervousness in Hilbert's voice as he began to lay out his vision for the future of mathematics. 'Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries?' To herald the new century, Hilbert challenged the audience with a list of twenty-three problems that he believed should set the course for the mathematical explorers of the twentieth century.

The ensuing decades saw many of the problems answered, and those who discovered the solutions make up an illustrious band of mathematicians known as 'the honours class'. It includes the likes of Kurt Gödel and Henri Poincaré, along with many other pioneers whose ideas have transformed the mathematical landscape. But there was one problem, the eighth on Hilbert's list, which looked as if it would survive the century without a champion: the Riemann Hypothesis.

Of all the challenges that Hilbert had set, the eighth had a special place in his heart. There is a German myth about Frederick Barbarossa, a much-loved German emperor who died during the Third Crusade. A legend grew that he was still alive, asleep in a cavern in the Kyffhäuser Mountains. He would awake only when Germany needed him. Somebody allegedly asked Hilbert, 'If you were to be revived like Barbarossa, after five hundred years, what would you do?' His reply: 'I would ask, "Has someone proved the Riemann Hypothesis?"'

As the twentieth century drew to a close, most mathematicians had resigned themselves to the fact that this jewel amongst all of Hilbert's problems was not only likely to outlive the century, but might still be unanswered when Hilbert awoke from his five-hundred-year slumber. He had stunned the first International Congress of the twentieth century with his revolutionary lecture full of the unknown. However, there turned out to be a surprise in store for those mathematicians who were planning to attend the last Congress of the century.

On April 7, 1997, computer screens across the mathematical world flashed up some extraordinary news. Posted on the website of the International Congress of Mathematicians that was to be held the following year in Berlin was an announcement that the Holy Grail of mathematics had finally been claimed. The Riemann Hypothesis had been proved. It was news that would have a profound effect. The Riemann Hypothesis is a problem which is central to the whole of mathematics. As they read their email, mathematicians were thrilling to the prospect of understanding one of the greatest mathematical mysteries.

The announcement came in a letter from Professor Enrico Bombieri. One could not have asked for a better, more respected source. Bombieri is one of the guardians of the Riemann Hypothesis and is based at the prestigious Institute for Advanced Study in Princeton, once home to Einstein and Gödel. He is very softly spoken, but mathematicians always listen carefully to anything he has to say.

Bombieri grew up in Italy, where his prosperous family's vineyards gave him a taste for the good things in life. He is fondly referred to by colleagues as 'the Mathematical Aristocrat'. In his youth he always cut a dashing figure at conferences in Europe, often arriving in a fancy sports car. Indeed, he was quite happy to fuel a rumour that he'd once come sixth in a twenty-four-hour rally in Italy. His successes on the mathematical circuit were more concrete and led to an invitation in the 1970s to go to Princeton, where he has remained ever since. He has replaced his enthusiasm for rallying with a passion for painting, especially portraits.

But it is the creative art of mathematics, and in particular the challenge of the Riemann Hypothesis, that gives Bombieri the greatest buzz. The Riemann Hypothesis had been an obsession for Bombieri ever since he first read about it at the precocious age of fifteen. He had always been fascinated by properties of numbers as he browsed through the mathematics books his father, an economist, had collected in his extensive library. The Riemann Hypothesis, he discovered, was regarded as the deepest and most fundamental problem in number theory. His passion for the problem was further fuelled when his father offered to buy him a Ferrari if he ever solved it - a desperate attempt on his father's part to stop Enrico driving his own model.

According to his email, Bombieri had been beaten to his prize. 'There are fantastic developments to Alain Connes's lecture at IAS last Wednesday,' Bombieri began. Several years previously, the mathematical world had been set alight by the news that Alain Connes had turned his attention to trying to crack the Riemann Hypothesis. Connes is one of the revolutionaries of the subject, a benign Robespierre of mathematics to Bombieri's Louis XVI. He is an extraordinarily charismatic figure whose fiery style is far from the image of the staid, awkward mathematician. He has the drive of a fanatic convinced of his world-view, and his lectures are mesmerising. Amongst his followers he has almost cult status ...

(Continues...)

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Excerpted from The Music of the Primes by Marcus du Sautoy
Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

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