Hypothesis finxit.(Stalking the Riemann Hypothesis)(Book Review)

Dan Rockmore Stalking the Riemann Hypothesis. Pantheon Books, 296 pages, $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, on the one hand, 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 cliches 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. 

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 ...