Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
448 pp., $18.00
The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
Farrar, Straus and Giroux, 2004
342 pp., $23.00
Three Prime Numbers Go into a Bar
Sabbagh is interested in what drives mathematicians (not the money, in case you were wondering). He seeks to describe flashes of mathematical insight and give a feel for the RH itself at a basic level, and he succeeds—but at a cost. Despite his intermittent attempts to portray high-level mathematicians as folks very much like you and me, his deep-seated attitude toward his subjects is captured well in the prologue: "It is given to them to see truths." Sabbagh's mathematicians are people, yes, but people who operate on a totally different plane when it comes to math, and often don't operate so well outside that rarefied realm. This picture isn't entirely wrong—Sabbagh's description of a typical math conference rings so true that, while reading the book at such a gathering, I thought maybe he was hiding behind the chalkboard, taking notes—but it's exaggerated. For example, his claim that mathematicians find thinking abstractly "a more satisfying activity than any of the other pleasures the world has to offer" needs a correction of "any" to "many."
Derbyshire's goals are entirely different, and so are his mathematicians. Prime Obsession is loosely written as a two-track book, one which goes into some detail about the math of the RH, the other following the history behind each advance along the way. As both tracks are following paper trails to some degree, the book is more grounded in reality. That's not to say Derbyshire shies from hyperbole; in the chapter "Turning the Golden Key," he refers to the mathematical step his prose has built up to for half the book by saying, "I simply cannot tell you how wonderful this result is."
But the historical view places such comments in context. We meet Bernhard Riemann, who proposed the now-famous hypothesis in a then little-noticed 1859 paper, in his milieu of the still-fragmented pre-Bismarck Germany. Like Sabbagh's de Branges (who appears in a single endnote here), Riemann seems "a rather sad and slightly pathetic character." Yet we also hear about his "very pious" Lutheranism and his devotion to his family. Each player in the narrative is thus portrayed, sympathetically and within the overall saga of the RH. If Sabbagh's book is a high-toned soap opera, this one is at least on the History Channel, maybe even PBS.  On the other hand, the story never grows boring, and it starts to prepare the reader for the in-depth mathematical treatment.
The mathematics does not totally drive the book (Derbyshire even suggests that the reader skip every other chapter once it gets too hairy). Still, it clearly excites him, and rightly so. I have never seen a treatment of the Riemann Hypothesis that builds up so clearly and with a writer's flair for suspense (Derbyshire is a novelist as well a popular historian of mathematics). That is the great strength of this "remarkable book," as Nobel laureate John Nash's jacket blurb suggests, and it is also the great weakness of an otherwise exceptional math popularization. Another blurb puts the dilemma best; the book "explains the hypothesis in ways understandable by ordinary mathematicians and even by laymen."  That locution—"ordinary mathematicians"—is nicely judged.
The early going is manageable, and some of the graphics are truly spectacular. On the other hand, even advanced undergraduates have difficulty understanding big-Oh notation and finite fields on the first try (and later), and ascent to the mathematical high point alluded to above requires considerable training and painstaking care. No one will go away without some level of understanding, but especially the last few math chapters would be well suited for a classroom setting, as is perhaps appropriate for a book published indirectly via the National Academy of Science.
The Riemann Hypothesis also tries to give some idea of what the RH is all about, but Sabbagh's tendency to over-elevate the (math-related) thoughts of mathematicians also infects his treatment of the math itself. He describes the mechanics of the statement at a level many readers will be able to understand; however, this leaves mysterious why Riemann should have made such a statement in the first place. This would be fine—that's not his goal, after all—if Sabbagh didn't at the same time leave behind some other readers completely. The "Toolkit" appendices will help readers who have forgotten the concepts involved, but they are not sufficient to teach a neophyte what a logarithm or an eigenvalue is.