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

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.

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, Goedel'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.


I'd give this book 6 stars if I could   May 18, 2006
Brian Michalowski (Seattle, WA USA)
7 out of 7 found this review helpful

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.





The best popular book on mathematics ever.   August 3, 2003
Michael Levitt (Stanford, CA, United States)
7 out of 8 found this review helpful

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!


CAUTION! This book may bring out your PRIMal passion!!   May 13, 2004
Stephen Pletko (London, Ontario, Canada)
7 out of 8 found this review helpful

+++++

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!

+++++



Fun and satisfying   January 10, 2004
Paul J. Papanek (Los Angeles, CA United States)
1 out of 1 found this review helpful

Clear, fun, with quite a bit of humor, and more mathematical detail than I thought I could possibly get through. I enjoyed reading this in tandem with the recent books on the Riemann hypothesis by Derbyshire and Sabbagh. If you liked either of those, this one will also be very satisfying for you.