Review of Prime Obsession

By James Alexander
(Professor of Mathematics at Case Western Reserve University)

“The Riemann Hypothesis” — it could be the title of a mystery thriller.  In fact, the hypothesis, R.H. to its aficionados, is a very deep assertion about basic properties of ordinary numbers.  Its roots go back to the earliest days of rational investigation, and some experts think it could remain a mystery for centuries.  Its story includes cryptic notes, missing manuscripts, close calls, false leads and dead ends and a host of characters.  It has a $1 million prize on its head, plus sempiternal professional fame for its solver.

R.H. concerns prime numbers, those atoms of ordinary multiplication, such as 5 or 11 or 1,223, that cannot be broken down — factored into a product of smaller numbers.  Most people are introduced to primes in early schooling, and the concept is stored away as a piece of trivia.  However, it does not take much numeracy to get hooked on primes.  Easily phrased questions can be quite sophisticated:  How to tell if a number is prime?  If it is not prime, how to factor it? How many primes are there? Is there a pattern to how the primes occur?

A method for the first question, arduously workable for numbers up to 10 digits or so, is named after the ancient polymath Eratosthenes.  For numbers that can be entered reasonably into a computer — say up to a few hundred digits — a definitive method was found by young Indian computer scientists last year.  Factoring numbers of this size is generally considered irreducibly difficult; that supposition is the basis of electronic security, from A.T.M. transactions through Internet purchases to terrorist communications.  But if a quantum computer can be built, the problem becomes manageable.  As for the number of primes, in Euclid's Elements there is an elegant proof that it is infinite.  And the fourth question above leads to R.H.

A pattern — that is the question.  A little investigation brings out a couple of phenomena.  Pick a number, say, your social security or bank account number, or those strung together.  First, the bigger it is, the less likely it is to be prime.  Second, whether it is prime seems inordinately random.  In 1859, Bernhard Riemann, a mathematician at the University of Gottingen, on his election to the Berlin Academy of Science delivered a short paper in which, expanding on ideas of his mentors, he profoundly rephrased these questions in a completely different form, and offhandedly posited what turns out to be the essence of the issue.  In the century and a half since, his supposition has assumed prime mathematical importance.  In fact, the likelihood that a given number is prime is roughly inversely proportional to its length; that much was teased out of Riemann's formulation in the 1890's.  The R.H. goes after the “roughly” — the inordinately random part — and implies it is as regular as could be hoped.

Riemann's formulation can be visualized as a landscape extending off in all directions.  In most aspects, the terrain is boringly uninteresting, but along one critical direction it has an infinite series of dips, called zeroes, and these zeroes code all the information about prime patterns.  R.H. asserts Riemann's zeroes lie in a perfectly straight line.  By computer computation, the first 10 billion or so do.  The zeroes are harmonics of the primes.  A musical instrument, unlike a tuning fork, does not produce a pure note but rather an infinitely rich collection of harmonics that make, for example, a clarinet's sound different from a violin's.  Each zero specifies a harmonic of the primes.  The assertion that all the zeroes are perfectly aligned implies that no prime harmonic dominates the others; they are all in a delicate balance.

Occasionally people have claimed to conquer R.H., but all claims have collapsed.  And R.H. has survived — sometimes by only a hair — all attempts to find a vagrant zero. Riemann himself never publicly returned to the subject, but his successors have found, in private scribbled notes he left behind, hard evidence that he knew more than he said, and tantalizing hints that he felt he had the final answer.  The pieces have never all come together.  Investigations have spun in many directions.  In 1972 a serendipitous connection was made with quantum physics.  At times money has been bet on R.H., and one can get action on either side.

Three new books circle around the Riemann Hypothesis, explaining what it means and how it came to be, and recount stories about people involved with it.  Although they necessarily overlap, they are quite distinctively presented.

John Derbyshire is a self-described generalist.  He is interested in the hypothesis itself.  His Prime Obsession alternates between brief biographical sketches of some of the principal players and an exegesis of Riemann's paper.  A reader phobic about displayed formulas will turn from this book.  Derbyshire attempts to walk the reader slowly up the mathematical slopes — only late in his book does he timorously introduce a bit of calculus.  However, a reader not willing to go with the mathematics will miss Derbyshire's intent.  Conversely, the reader willing to work through Derbyshire's presentation will understand something of Riemann's insights.

Marcus du Sautoy's Music of the Primes contains virtually no formulas; yet in another sense it is the most mathematical of the three.  He 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.  He tells insider stories — probably mostly true.  He free-associates, casually introduces ideas and people peripheral to the central topic and then, like a jazz musician, cycles back to the main theme.  In the other books, the reader is taken along the authors' personal roads of mathematical discovery.  Du Sautoy starts well along the road, and his paths are different byways.

Karl Sabbagh is a producer and writer for the BBC, and he brings that perspective to his writing. He tells a different set of stories, sometimes looking at mathematics from the outside. His book is full of people talking about R.H. — what it means and how they think about it — and about themselves and their colleagues.  He writes about R.H. itself, but late in his book notes, “You know almost nothing about the Riemann Hypothesis compared with what there is to know.”  His book could easily have been a TV documentary.  Derbyshire uses “Obsession” in his title.  Du Sautoy writes that John Nash illustrates “the dangers of mathematical obsession” — an unfortunate and unwarranted assertion.  Louis de Branges of Purdue University is unmentioned by Derbyshire and du Sautoy but a significant figure in Sabbagh's book; perhaps his “single-minded focus” is closer to mathematical obsession.

All three authors want us to understand how Riemann transformed questions about primes into ones about dips in his landscape — from the steps of counting numbers to the continuum of measuring numbers, as Derbyshire puts it.  Sabbagh, with Riemann, calls the transformation “Euler's product,” referring to the Swiss mathematician Leonhard Euler, and devotes an explanatory “toolkit” to it.  Derbyshire calls it the “golden key”; du Sautoy calls it a “wormhole” between mathematical universes or Alice's looking glass.  To appreciate this transmutation is to grasp the raison d'etre of the books.

A few years ago, a review of a biography of the mathematician Paul Erdős included a suggestion that there was some mathematics in the book but it could be easily skipped.  Just so, I suppose it is possible for a biography of James Joyce to ignore his writing.

Rather, take this: some cultural landmarks are near where I work — the Cleveland Symphony, the Museum of Art and others.  Performing and visual arts are part of the human experience and, although I cannot claim scholarly expertise, with some application I can appreciate the performances and exhibits.  Mathematics at this level is itself a fine art, with its own aesthetics, culture, context and history.  With the Riemann Hypothesis as a motif, these books offer for different tastes three complementary entrees to this culture.