The Music of the Primes

The Riemann Hypothesis is one of those problems in mathematics that is inobvious and difficult to explain to someone without a mathematical background. Put baldly, it's the hypothesis that all non-trivial zeroes of the Riemann zeta function lie on the critical line at one-half. This is almost entirely unenlightening. To understand why this is one of the most famous problems in mathematics, and why it's worth writing a 300 page book about it, requires a great deal of background and explanation. To aim that book at a non-mathematical audience requires a lot of analogies, a lot of careful description, and skill for presenting detail without making the book overwhelming or dry.

This is the second book about the Riemann Hypothesis that I've read. Dr. Riemann's Zeros also provided background and explanation of the problem, but it was primarily concerned with the personalities and current approaches to the problem. Du Sautoy takes a much different approach than Sabbagh: The Music of the Primes is a history. It is specifically the history of attempts to analyze the properties of prime numbers, but along the way it tells some of the history of number theory in general. It isn't until 80 pages in that Riemann even makes an appearance.

Readers of my previous reviews will know that George Gamow's One Two Three... Infinity made a huge impression on me, particularly the early chapter on number theory. As an expansion of that chapter, The Music of the Primes is a dream. Du Sautoy not only covers all of the ground that Gamow only mentions. He also adds many new players (Fourier, Hilbert, Hardy, Littlewood, Ramanujan, Gödel, Erdős, Turing, Shamir, Rivest, Adleman, and others) and takes side journeys through mini-biographies and summaries of the work of many of the mathematicians who have touched the Riemann Hypothesis or prime numbers. Among the digressions is the story of the development of RSA public key cryptography and one of the best explanations of Gödel's Incompleteness Theorem that I've ever read, not to mention a history of the mathematics department at the University of Göttingen. This additional humanizing of mathematicians, putting them into a social context and looking at the psychological impact of major developments in the field, is one of the things that makes The Music of the Primes such compelling reading.

The other is that du Sautoy is exceptionally good at explaining mathematical concepts in simple ways. Here, I do always have to warn that I have a reasonably good background in mathematics, including some graduate-level classes, so although it's been fifteen years since I've used that knowledge, I'm not easily out of my depth in a popularization. But that said, I think du Sautoy's explanations would be clear to anyone who got as far as infinite sums, and even that might not be needed. And despite that clarity, he goes much deeper into the details of what has been proved (and how) than Sabbagh does.

A brief overview: Riemann's Hypothesis is important because of prime numbers. Prime numbers, previously of modest practical import despite being at the heart of number theory, have acquired new substantial importance by forming the heart of most public-key cryptography. Gauss started the ball rolling on the search for deeper understanding of prime numbers by discovering a startling connection between the number of prime numbers less than or equal to N and the natural logarithm of N. Euler contributed by finding a way to rewrite the zeta function, an infinite sum, as an infinite product of terms formed from prime numbers, thus linking the two fundamental operations of arithmetic via the primes. Riemann put them both together by proving that his expansion of the zeta function, when fed imaginary numbers, provided a much better estimate of the number of primes. And, going further, he found that he could construct a sort of wave equation from each zero of the Riemann zeta function and, by adding those waves into the estimate in a way similar to Fourier synthesis, he could produce a perfectly accurate description of prime numbers.

If that involved too many concepts at once, rest assured that du Sautoy goes over each one in depth, accompanied by biographical sketches of the mathematicians who worked on it and sometimes summaries of their other work. He then goes on to explain why this result means that the Riemann Hypothesis has profound implications for our understanding of the bounds and nature of the distribution of prime numbers and why a proof would open doors to better understanding how numbers are constructed. And this is just the first third of the book. The rest discusses the growing interest in the Riemann Hypothesis, the loss of Riemann's papers, the Hilbert problems, and numerous other fascinating bits of both mathematics and mathematical history.

One example that sticks in my mind is the proof that Gauss's original estimate for the number of primes does not always overestimate the number of primes as he believed; instead, it begins to underestimate the number of primes by at least the point of the Skewes number. However, the Skewes number (particularly in the original proof) is a number so large that it's vastly larger than any number that would be possible to calculate. I'd not heard about this before reading the book, and du Sautoy excels at explaining not only the tidbit of knowledge but why it's so mind-boggling that a property of numbers would change that far outside the reaches of any conceivable computation. Another fascinating tidbit is that we now have a formula which produces every possible prime number, but it's an extremely unsatisfying formula formed by converting a Turing machine to a polynomial, varying 26 variables, and throwing out all negative results. We can produce primes on demand, but only through the mathematical equivalent of monkeys pounding on 26 keyboards.

I could go on; The Music of the Primes is a treasure trove. I enjoyed Dr. Riemann's Zeros, but this is far and away a better book. Structuring it as a history with digressions succeeds marvellously, the stories of the mathematicians gives the book a narrative backbone and human interest to bridge the mathematics, and it is delightfully full of side excursions and related results that made me think. I do wish that du Sautoy would have sketched the outlines of more of the proofs, since he's far better at explanation than most of the resources available on-line. This book was a little more free of difficult material in places, particularly towards the end, than I would have preferred. But that's a minor quibble and probably improves the book for others.

You have to have a certain bent of mind to enjoy this sort of thing, and despite the history of mathematicians you probably won't love this book unless you also love math. But it's pure delight for anyone who likes number theory and the best popularization of serious mathematics that I've ever read. If my review above didn't make your eyes glaze over, search this one out.

I leave you with this startling idea, also mentioned by du Sautoy, and another good example of the surprising bits of mathematics he explains so clearly: if one can prove that the Riemann Hypothesis is impossible to prove one way or the other using our axioms of mathematics, something that Gödel's Incompleteness Theorem says is possible and which has been proven for other questions, then one will have proven the Riemann Hypothesis to be true! If the hypothesis is false, then there is a zero off of the critical line. If there is a zero off of the critical line, it would be possible to discover the input to the Riemann zeta function that yields that zero, and hence disprove the hypothesis. Hence, if the hypothesis cannot be proven either way, there must not be a zero off of the critical line, and therefore it must be true.

Rating: 9 out of 10

Reviewed: 2009-12-23