One Man's Zeta Jones

National Review, July 28, 2003 by David Gelernter

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, by John Derbyshire (Joseph Henry, 304 pp., $27.95)

This is a striking and brilliant book, in many ways the most ambitious science-for-the-public attempt I have ever read. John Derbyshire undertakes a task which is (we are more or less convinced by the end) impossible, and yet the book succeeds, and at its best it is beautiful. It reads as if it were written not merely by a mathematics scholar but by a first-rate novelist -- and that is what Derbyshire is.

An unmathematical reader might have difficulty in following it all but not in reading it. If you can't stretch your concentration far and wide enough to cover the whole thing, there are nonetheless so many provoking and illuminating observations along the way that you will be scooped up and carried along whether or not you ever come to terms with the big picture.

Derbyshire sets out to explain "the greatest unsolved problem in mathematics," the Riemann hypothesis. It is Derbyshire's bad luck that, over the last generation, two other problems that used to head up the Greatest Unsolved chart have been disposed of. Those two -- the Four Color Theorem and the ever-popular Fermat's Last -- each had the winsome appeal of a small cuddly animal. Four colors are sufficient (says the Four Color Theorem) to let you tint any ordinary black-and-white map in such a way that no two adjacent countries or regions have the same color. Sounds simple and is. Anyone can heft it and (for that matter) cuddle it: Nearly every math or science student figured he could discover a proof, and killed an enjoyable few days trying. Such unsolved problems were no mere spectator sports. Everyone could play.

But today, it is Riemann's turn atop the list. Derbyshire shows us that the Riemann hypothesis is a broad, deep, and fascinating topic -- but the hypothesis states (I regret to inform the reader) that "All non-trivial zeros of the zeta function have real part one-half." Which might possibly not strike you as the sexiest proposition you ever saw.

It takes a fair amount of mathematics even to say what the zeta function is, let alone why anyone should care. But Derbyshire sets out to explain the hypothesis, how it relates to a deep and fascinating fact about prime numbers, and how it connects to many other far-flung parts of mathematics and physics. Prime numbers are the heart of the story, and they (if not the hypothesis itself) are easily grasped. As you count upwards from 1, prime numbers -- which are divisible only by themselves and 1 (2, 3, 5, 7, 11, and so on) -- keep showing up, but less and less often. They grow ever scarcer, yet never peter out entirely. Between 1 and 100 there are 25, between 901 and 1,000 only 14; in the last 100-count before one trillion, a mere 4.

In 1859 the great mathematician Bernhard Riemann published a paper about the Prime Number Theorem -- which quantifies the Petering-Out of the Primes -- and in the process invented "a mathematical object of great power and subtlety" . . . and then threw out a "casual, incidental guess" about this fine new object he had invented. The new object was the zeta function, and his guess came to be known as the Riemann hypothesis -- which has since become "an obsession" among mathematicians, says Derbyshire, "having resisted every attempt at proof or disproof. Indeed, the obsession is now stronger than ever."

The hypothesis makes a rich and fascinating topic because it encapsulates so much and such varied mathematics, and has such wide-ranging implications. It is a mathematical opal winking and shimmering with a million colors; contemplate this one gem long and carefully enough and you will see whole worlds without moving from the spot. But (of course) this richness is exactly what makes the topic a potential killer to write about, especially (but not only) if your book is for laymen.

To explain the hypothesis, Derbyshire force-marches his readers through the fields of functions and limits and infinite series and natural logarithms, and across the complex plane; he allows us a refreshing quick dip into calculus and then (onward!) to "a little algebra" (but not too little), chaos theory, "the vis viva equation familiar to all students of elementary celestial mechanics," and beyond.

"A Little Algebra" is a chapter that shows the method's successes and failures. Derbyshire's discussion of field theory (algebra over specially restricted sets of numbers or quasi-numbers) is quick but clear, bracing, and fairly easy to follow. The author then spots himself eight pages for "operators" (which have to do with matrices, etc.) -- and leaves a broken heap of dying half-explanations behind him as he marches inexorably forward. Apologies to the "mathematically fastidious," instructions to go look it up "in any decent algebra textbook," "suffice it to say that they exist," "I'm afraid I can't explain just how you find . . ." -- these statements all crop up within a single two-page stretch. They are symptoms of an author's having tried manfully to run up a down escalator and failed.