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The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics

Karl Sabbagh
Publisher: 
Farrar, Strauss and Giroux
Publication Date: 
2003
Number of Pages: 
304
Format: 
Hardcover
Price: 
25.00
ISBN: 
0374250073
Category: 
General
[Reviewed by
S. W. Graham
, on
08/20/2003
]

Within the past few months, we have seen the publication of three popular books on the Riemann Hypothesis (RH). I have reviewed the book by Derbyshire for MAA Online. I have not read the book by du Sautoy, but it too is being reviewed on MAA Online.

The books by Derbyshire and Sabbagh focus on different aspects of the Riemann Hypothesis story. Derbyshire writes at length about the mathematics behind the Riemann zeta-function, and he makes a considerable effort to explain it to a non-expert reader. Sabbagh's discussion of the mathematics is fairly cursory, though he does have some nice graphs of the zeta-function in Chapter 5. Derbyshire writes extensively about Riemann's life, and he puts it in the context of the political and educational landscape of 19th century Europe. Sabbagh has little discussion of Riemann's life or times. Instead, he concentrates on contemporary mathematicians, and he attempts to explain what contemporary mathematical research is like.

Sabbagh has done interviews with over twenty mathematicians who work in areas connected to the Riemann Hypothesis, and he includes extensive quotes from the interviews. He describes his approach in anthropological terms — "I am describing a remote tribe whose customs and language are unfamiliar to the reader, but whom I understand enough to convey something of their inner and outer lives." We learn, for example, that most of them decided to become mathematicians very early in life. We also learn their opinions on whether or not RH is true — most of them feel that it is true, though a few are skeptical. No one wants to believe that it is undecidable. We learn their feelings about the prospects for a proof. For example, "[Henryk] Iwaniec doesn't believe anyone today has a strong or convincing program. 'Never mind a complete proof, but a program, a direction to go in.' But he is certain of one thing: 'Mother Nature has such beautiful harmonies, so you couldn't say something like that is false.'"

Overall, Sabbagh imparts a reasonably accurate view of what mathematicians are like, although he probably overemphasizes some of the stranger aspects of our behaviour. The joys and frustrations of mathematical research are well portrayed. There are many good quotes in the book, and it will probably be widely quoted. Indeed, one can already find numerous passages from the book on the internet.

Of the mathematicians interviewed for the book, the one most prominently featured is Louis de Branges of Purdue University. De Branges, is of course, well known for his proof of the Bieberbach Conjecture in 1984, and he has been working on the Riemann Hypothesis ever since. He has several times announced a proof of RH, only to retract it later. Sabbagh acknowledges that other number theorists are skeptical about de Branges work, but their doubts did not dissuade him. Sabbagh writes "... I chose de Branges for the simple reason that he told me he was putting the final touches to a proof. After getting over my initial excitement, it took only two or three phone calls to realize that no one else took such a statement seriously. ... their skepticism made me want to find out more about de Branges, as a way of beginning to understand what mathematicians do when they are trying to prove a major hypothesis."

Among other things, Sabbagh gives excerpts from reviews of de Branges' declined National Science Foundation (NSF) proposal, submitted in 2002. The reviews are fairly negative, but Sabbagh presents them sympathetically, as if to argue that the NSF and its reviewers are trying to stifle the efforts of someone who is about to make a breakthrough on the Riemann Hypothesis. However, de Branges has enjoyed considerable past support from the NSF. Specifically, he was supported by the NSF continuously from 1984 through 1998, and again from 2000 through 2002. In all, the NSF has awarded six grants totaling $459,279 for the work of de Branges on the Riemann Hypothesis. (This information is publicly available at the NSF Fastlane web site.)

As a former program director at NSF, I know that program directors there will take a chance on risky proposals that attack long standing important unsolved problems, particularly if the principal investigator has a good track record. De Branges is one example of someone who has been supported for a high-risk proposal on some long unsolved problem; there are other examples. It is appropriate that such investigators be given some time to investigate a risky approach. On the other hand, it is not surprising that someone with sixteen years of support and little to show for it would be turned down. The conventional wisdom is that de Branges' approach will not work. But if de Branges does succeed, and the conventional wisdom turns out to be wrong, then Sabbagh's book will look quite prophetic.


References:

John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press, 2003.

Marcus du Sautoy, The Music of the Primes, Harper Collins, 2003.


S.W. Graham is currently a Professor of Mathematics at Central Michigan University. From 1995 to 1998, he was a program director in the Algebra and Number Theory Program at the National Science Foundation. Before that, he held faculty positions at Michigan Tech, the University of Texas, and California Institute of Technology. He received his Ph. D. at the University of Michigan under the direction of Hugh Montgomery in 1977. He can be reached at sidney.w.graham@cmich.edu.

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