»  The New Criterion

November 2005

  Hypothesis Finxit

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Stalking the Riemann Hypothesis
by Dan Rockmore
Pantheon Books; 296 pp. $25

The author of a pop-math book must decide, before he sets finger on keyboard, how much he is going to demand of his readers in the way of willingness to engage with actual mathematics. As is often the case in writing, what is easier for the author is more difficult for the reader, and vice versa. If you decide to press your reader's nose to the grindstone, you can present pages of standard equations and calculations, and guide him through them. This is hard for the reader, but easier for the author, who has only to regurgitate some well-established mathematical clichés and supply connecting prose. If, on the other hand, you seek to hold the attention of an ordinary educated person without taxing his mathematical knowledge too much, then you must cloak your mathematics with clever metaphors, and link the metaphors together in such a way that your mathematical narrative becomes an elaborate allegory expressed in ordinary language. That is very difficult to do. It may, in fact, be impossible. At any rate, I have seen no examples that struck me as really successful. Possibly I am hypercritical; pop-math books, including some with barely an equation or graph to be seen in their pages, sell quite well.

Dan Rockmore's book about the Riemann Hypothesis (hereinunder "the RH," in accordance with common usage among mathematicians) takes a middle path, falling back on metaphors for the more abstruse concepts presented, but doing as much as it can with tables and diagrams. It contains only two equations. Rockmore humanizes his narrative with good coverage of the background history of the RH, and of the personalities who have engaged with it. Altogether I think he has balanced his material very well, and given a fluent and readable account of this greatest of all "open" problems. The metaphors he has used to cover the more abstruse areas of the subject are not original, but they are well presented. For example, Rockmore relies on Sir Michael Berry's "music of the primes" analogy — Sir Michael actually played the "music" at a 1996 conference — to explain the relevance of the RH to the distribution of prime numbers.

The elemental primal resonances are described by very special complex numbers, and this is where Riemann's zeta function makes its magical appearance. For the complex numbers that delineate the fundamental tones whose symphony is the accumulation of primes are precisely those complex numbers that bring Riemann's zeta function to zero. They are the tunings on the dial of the Riemannian PDA [prime distribution analyzer] that cause its readout to flatline. These settings are called the zeros of Riemann's zeta function …

[Author's italics.] So there you are. How much understanding this will convey to a reader not schooled in the subtleties of complex variable theory, I cannot say, but it seems to me to do as much as can be done with the metaphorical method.

The RH, now the most challenging unresolved conundrum in mathematics, dwells in the realm of pure theory. This presents additional difficulties to the pop-math expositor. It is all very well (the ordinary reader will say) to speak of the aesthetic and intellectual pleasures of higher mathematics, but what use is it? The RH is, on the face of it, no use at all. If it could be resolved — proved true, or proved false — the only direct result would be the clearing-up of some fine points about the way prime numbers are distributed amongst all other numbers. No-one would build a better bridge or bake a better cookie in consequence. This is the purest of pure mathematics. Some math propagandists have argued that a resolution of the RH would have consequences for modern methods of cryptography, which rely on certain properties of the primes. I am not myself convinced by this. Neither, to judge from his minimal coverage of the topic, is Rockmore.

Much more thought-provoking is the connection between the RH and certain models for the behavior of systems of subatomic particles. Thirty years after this connection was first spotted, there is still no real understanding of what, if anything, it signifies. If the largest claims are correct, the RH represents a particularly startling instance of what Eugene Wigner, in a 1960 essay, called "the unreasonable effectiveness of mathematics in the natural sciences." Why on earth should the behavior of subatomic particles have anything to do with the distribution of prime numbers? It is still possible, though, that this entire line of inquiry is a dead end.

More probably what we have glimpsed there is some very deep principle underlying mathematical statistics. The truths uncovered by modern science, from nuclear physics to evolutionary biology, are more and more of a statistical nature. The ubiquity of the famous bell curve, a purely mathematical construction, illustrates the point. Researches during the past forty years have found constructions nearly as ubiquitous, but far subtler and more mysterious, whose applicability spans many mathematical and scientific disciplines. This is an important associated field of inquiry, and I think the author was right to give an entire chapter to it, though I am not sure that his account will leave a reader understanding the relevance of the RH to these speculations, other than that the properties of random matrices somehow provide a key.

Bernhard Riemann first stated his vexing Hypothesis in a paper he presented to the Berlin Academy in 1859. It was a throw-away remark, incidental to the purpose of his paper (which was, to develop a precise mathematical formula for the number of primes you will find in any range). We shall soon be at the 150th anniversary of that paper, and interest in the RH will no doubt intensify, as it did in 2000, when the RH was included in a list of seven "millennium problems," for each of which a $1 million prize was offered by a private research institute in Cambridge, Mass. It is a good thing for mathematics, and for our intellectual culture at large, that this wonderful problem, with all its strangely suggestive connections to so many widely separated fields of knowledge, should be brought to the attention of curious non-mathematicians. Dan Rockmore has produced a fine exposition here, worth the attention of anyone who wants to get acquainted with this most challenging, most mysterious of all higher-mathematical puzzles.