Review of Prime Obsession

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Prime Obsession

by John Derbyshire

WASHINGTON, DC, JOSEPH HENRY PRESS, 2003, $27.95, ISBN 0-309-08549-7

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The Music of the Primes

by Marcus du Sautoy

NEW YORK, HARPER COLLINS, 2003, $24.95, ISBN 0-06-621070-4

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The Riemann Hypothesis

by Karl Sabbagh

NEW YORK, FARRAR, STRAUSS AND GIROUX, 2003, $25.00, ISBN 0-374-25007-3

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REVIEWED BY HAROLD M. EDWARDS

The nearly simultaneous publication of three books for the general public about the Riemann hypothesis (hereinafter referred to as RH) can probably be explained by the million-dollar prize offered by the Clay Mathematics Institute for the resolution of RH (large sums of money evoke interest) and by the many books that were sold to the general public about Fermat's last theorem in the wake of Wiles's proof (selling books is the goal of publishing). Whatever the reason for this sudden flood of interest in one of the frontiers of pure mathematics, it is a welcome, if surprising, phenomenon.

Mathematicians are probably the worst people to review such books. An architect I once met pleased me by telling me how he had become convinced of the power and beauty of mathematics by reading a certain popular book on mathematics that he named. I was so gratified by this deviation from the usual "I was never any good at math" that I rushed to the library to see the book. My disappointment was great. To me, it was full of dubious assertions, exaggerations, oversimplified history, and explanations of mathematical ideas that could impart no understanding other than false understanding.  But,  as  the  architect plainly demonstrated by his own example, the book had achieved its goal brilliantly, at least for one reader.

Moreover, I have had the experience—and most mathematicians I have asked about it have had the same experience—of rereading a book for non-mathematicians that I had read in my youth and that I remembered as having inspired me, only to discover that it had many explanations I now found to be misleading at best and statements I now found to be downright wrong. Would I recommend the book to a young reader today? My own experience would say yes, but my judgment as a mathematician would say no.

These considerations have been on my mind as I pondered these books on RH. All three are quite well written, and I can easily imagine any one of them capturing the non-mathematical reader's fancy. And, overall, I think each presents a reasonably accurate picture of the history of RH and the present-day mathematicians who are working on it.  For that, the mathematical fraternity can thank all three authors. But, after all, I am a mathematician, and it is only as a mathematician that I can evaluate the books.

My lack of success over the years in explaining the irrationality of to reasonably able liberal arts students has left me without much hope that explanations of RH intended for interested non-mathematicians will succeed. In other words, I am among the "many people" who, according to the first sentence of Karl Sabbagh's Prologue, "would say that the task I am embarking on ... is futile." He defends his project by comparing it to anthropology and to "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."  Readers of the Mathematical Intelligencer, as members of that remote tribe,  will  be  interested  to  know whether the descriptions he provides are accurate and whether they illuminate our tribal culture. On both counts, I am unenthusiastic.

The best parts of the Sabbagh book are indeed the anthropological ones.  He tells who has worked on—or is working on—the Riemann hypothesis, how they became interested in mathematics and in this particular problem, how they view their chances for success, and so forth. But what makes us a tribe is our peculiar culture, and there is no way to describe the interactions of key members of the tribe without going into the substance of our culture.  On this, Sabbagh is an unreliable guide.

For example, on p. 41 (page numbers refer to the American edition—the original English edition is more compact, so for example, this passage is on page 33 of that version) he says: "So, calculating the value of the sum , which Riemann believed was possible but couldn't say so for certain, would result in a totally accurate number for the number of primes less than n."  Well, calculating the value of  is certainly possible when the real part of s is greater than 1, but Sabbagh does not say which particular values of s will be needed to produce his "totally accurate number." (Later in the book, complex numbers are introduced, and on the next-to-last page analytic continuation is mentioned in passing, but at this point  is far from being the same thing as .) This "totally accurate number" must refer to Riemann's explicit formula for , which Sabbagh seems to believe (see also the end of Chapter 1) depends on RH; but in fact what is needed for this formula is not the evaluation of  for one or more values of s but a knowledge of the zeros of  in the critical strip; the formula is valid whether or not the zeros are on the critical line. This misapprehension about the meaning of RH probably underlies his answer, at the end of his Prologue, to the question, "Why is it [RH] so important? ... A proof... would... tell mathematicians a huge amount about an important class of numbers—the prime numbers, which dominate the field of pure mathematics." The notion that such a goal accounts for the fascination of RH is a profound misunderstanding of our tribal  culture, like  believing  mountaineers want to climb Mount Everest in order to get somewhere.

For another example, he often speculates about who might or might not prove the Riemann hypothesis. On p.219, Martin Huxley is said to have "both the desire and the ability to prove the Riemann Hypothesis." On p. 240, we are told that "many . . . feel that if anyone is going to prove the Riemann Hypothesis, it will be [Alain Connes]." And not only is there speculation that Louis de Branges might be the one to prove RH, the book includes an Appendix by de Branges with the title "De Branges's Proof." This isn't the way research in mathematics goes. Perhaps in other fields that require expensive equipment one might, to a limited extent, predict where the next breakthrough might occur, but in mathematics any attempt to predict whether there will be a proof any time soon, much less what shape it might take or who might devise it, is completely foolish.

In this connection, Sabbagh gives us an interesting pair of speculations:  Henryk Iwaniec (p. 36) says, "I'm only worried that what may happen is that a proof will be given by somebody and I will be unable to understand it," while Alain Connes (p. 263) worries about something quite different: "It would be a tragedy if it just needed a trick to prove it." Different as these concerns seem, I suspect that most mathematicians sympathize with both. Note that neither has anything to do with "learning a huge amount about an important class of numbers."

Another of the authors, Marcus du Sautoy, is a professional mathematician, so we should expect his statements to be correct, but I am puzzled by his statement (p. 11) that "a proof of the Riemann Hypothesis would mean that mathematicians could use a very fast procedure guaranteed to locate a prime number with, say, a hundred digits or any other number of digits you care to choose." He goes on to relate this to RSA cryptography, clearly implying that RH would have some practical significance for cryptography, but I doubt that this is the case. I suspect, rather, that he feels the general reader must be given some reason for the significance of this million-dollar question in mathematics, but that

the real reason depends on aspects of our tribal culture that are too difficult to explain to the general reader. (Perhaps I am wrong; the supposed connection is again mentioned on page 243.) Similarly, on page 12 he says, "The security of RSA depends on our inability to  answer basic  questions about prime numbers," but I thought it depended on our inability to factor large numbers. In fact, I thought the practicality of RSA depended on the disparity—in practice, primality testing is easy; factoring, hard.

His statement on page 5 that "Mastering these building blocks [primes] offers the mathematician the hope of discovering new ways of charting a course through the vast complexities of the mathematical world" puzzles me in a different way. Whatever could it mean? To my taste, this statement, and much else in the du Sautoy book, sounds too much like empty enthusiasm, razzle-dazzle meant to impress non-mathematicians  not  with  substance—because substantial mathematics is beyond their ken—but with fanfares and flourishes.

Du Sautoy touches on a clash of cultures within mathematics that is seldom revealed to outsiders and that might hold some interest for Sabbagh and others interested in the anthropology of our tribe. In his last chapter he sketches in very laudatory terms the career of Alexander Grothendieck; "Grothendieck's new language of geometry and algebra saw the creation of a whole new dialectic which allowed mathematicians to articulate ideas which were previously inexpressible" (p. 300). Then he goes on to say of the brave new world of the Grothendieckists that "Even [Andre] Weil was rather disconcerted by Grothendieck's new abstract world," and, even more baldly, quotes Carl Ludwig Siegel as saying, "I was disgusted with the way in which my own contribution to the subject had been disfigured and made unintelligible," and Atle Selberg as saying, "My thought was that such lectures were never given in earlier times. I said to someone after the lecture a thought which had come into my mind: if wishes were horses, then beggars [could] ride."  Disagreements at the highest levels of mathematics are extremely interesting, and I applaud du Sautoy for bringing them into the open, although he does not pursue the subject.

I do not applaud, on the other hand, his description of the relation between RH and mental illness. In his last chapter, he says "Grothendieck is not the only mathematician who has gone crazy trying to prove the Riemann Hypothesis," as an introduction to a paragraph about John Nash. "Grothendieck and Nash illustrate the dangers of mathematical obsession," he concludes, but mathematical obsession, whether it is with RH or the continuum hypothesis, is surely a symptom, not a cause. Our tribe may have a stronger than average association with madness that deserves to be addressed, but, if so, it deserves to be addressed with more seriousness than to talk about going crazy trying to prove the Riemann hypothesis.

As a sometime historian of mathematics, I am dismayed by du Sautoy's failure to cite a single one of his historical sources. On page 104 he tells of "several drafts" of a letter he says Riemann was writing to Chebyshev about "his own progress" in the investigation of the prime number theorem. In my 1974 book Riemann's Zeta Function I published a jotting from Riemann's helter-skelter notes showing that he was aware of Chebyshev's existence; if there is more evidence than this of a Riemann-Chebyshev connection, I do not know about it. The account of Siegel's military history (p. 148) differs substantially from the one given by Benjamin Yandell in The Honors Class, and, since Yandell names his sources, I believe his. I hope most readers will realize that no sources could possibly

support such statements as "[Pythagoras]  filled an urn with water and banged it with a hammer to produce a note" and so forth on p. 77, but many readers will not. Surely I am not the only reader who wants to know what lies behind the surprising reference (p.128) to "Gauss and Einstein's belief that space was indeed curved and non-Euclidean." That Gauss might have considered the possibility of non-Euclidean physical space is plausible enough, but that he believed it? The propagation of unchecked and uncheckable anecdotes about the history of mathematics is a form of pollution to be combatted. An occasional tall tale, with appropriate caveats, can certainly be used to spice up the exposition from time to time, but when no sources are ever given for anything, such tales become an unacceptable norm.

Another feature of du Sautoy's writing is his habit of introducing a private phrase to describe something and forever calling it by his new name rather than the one used by everyone else. For example, he says on page 20 that "One of Gauss's greatest early contributions was the invention of the clock calculator." He goes on to explain what he means—modular arithmetic, of course, the "clock" being a reference to arithmetic mod 12—but thereafter there is no modular arithmetic, only "clock calculators" as in "That is because the calculations will be done not on a conventional calculator, but on one of Gauss's clock calculators" (p. 234, dealing with RSA). Similarly, zeros of  are first described as "points at sea level in the zeta landscape" (p. 89) and are called that for the remainder of the book. On page 79, rather than saying that the harmonic series diverges, he says it will "spiral off to infinity"—an odd way to describe gradual increase without bound—and thereafter series never diverge but "spiral off to infinity."  The line where the real part of s is ½ is not the critical line, it is "Riemann's magic ley line" (p. 98) or "Riemann's ley line." I have not found a definition of "ley" in any American dictionary that fits this use; it is apparently a term used in British surveying.

But du Sautoy and Sabbagh were not writing for mathematicians. It may well be that the general readers they have in mind will be intrigued and gratified by their descriptions of mathematics and mathematicians related to RH. Certainly there is amusement to be found in these books, and even mathematicians will find many interesting things in them if they are not too distracted by questionable formulations and implausible anecdotes. No harm is done as long as cranks are not encouraged and as long as genuinely inquiring minds are not put off when some of the purported explanations do not seem to make sense.

The goal of the third book, the one by John Derbyshire, appears to be different from that of the other two. He writes in his Prologue of "a general readership" (p. xii), but I think he is unduly optimistic. He mentions, for example, that he expects his readers to understand basic algebra, such as the fact that S = 1 + xS  becomes  S = 1/(1 - x) when rearranged. Certainly anyone setting out to understand RH must be comfortable with this rearrangement, but, in the first place, I suspect that more educated adults than we like to imagine would not be comfortable with it, and, in the second place, more mathematical sophistication and ability is needed than this example suggests. Perhaps Derbyshire set out to write a book for the general reader, but as it developed I think his goal had to change.

No matter! He has written a wonderful book. He does not fudge the mathematics, which will make parts of it hard going for most non-mathematicians, but for the most important audience of non-mathematicians—those young ones who might consider becoming mathematicians—it will be a great resource and inspiration. And for mathematicians and readers with a fair amount of mathematical sophistication, it is a book that will inspire, inform, and entertain. If you believe as I do that RH for the general reader is a futile project, you will agree that Derbyshire made the right choices. The copyright page states that the publisher, the Joseph Henry Press, "was created with the goal of making books on science, technology, and health more widely available to professionals and the public," a goal that is admirably served by this book.

(Full disclosure: Derbyshire names me in his acknowledgments and mentions me a few times in the book. I met him very briefly at the conference on RH held at NYU in 2002, and at that time I gave him a copy of my book, but I don't recall anything else requiring acknowledgment. Of course, I have made every effort to base my opinion of his book on the book alone.)

Even experts on RH will enjoy this book and learn from it, and I would encourage all readers of the Mathematical Intelligencer to try it. It is interestingly and  skillfully written,  and  it approaches many aspects of the subject in imaginative and thought-provoking ways. For a quick probe, you might try reading pages 90-92. There you will find discussions of the contrast between measuring and counting (as he describes it, numbers legato and numbers staccato), Gauss's attitude toward Fermat's last theorem, Mallory's reason for wanting to climb Mount Everest, the rise of the Germans in 19th-century mathematics and how it may have been related to the Napoleonic wars and the Congress of Vienna—as well as a passing mention of Larry, Curly, and Moe.  If that rushed summary suggests that the writing is contrived or precious or pretentious, the fault is mine.  To my taste it is always down-to-earth and treats its topics in natural and appropriate, but interesting, ways.

Naturally I have my disagreements and cavils with the book, but it is remarkable to me how few they are when I consider how dense with information and opinions the book is. The peasant-pheasant story about Peter the Great on p. 56 should have been tossed out in the revision process.  Derbyshire  does an admirable job of keeping the  calculus in the book to a minimum (he tells us in the Prologue that his original goal was to have no calculus at all, but that this goal "proved a tad over-optimistic"), but my alarm bells go off when, on p. 112 and again on p. 113, he describes a definite integral as an "area under a function." I gather that, competent as his mathematics is, he has never taught calculus and had to deal with students who persistently confuse functions with their graphs.

I am most disturbed by his statement about the formula  that "Gauss is supposed to have said—and I wouldn't put it past him—that if this was not immediately apparent to you on being told it, you would never be a first-class mathematician," not only because I question the attribution of such a statement to Gauss and no source is given, but mainly because it strikes me as a terrible thing to say to a young student. One's reaction to  must be awe, not "oh, yes, of course!" If you tell me it was immediately apparent to you when you first saw it I will think you are a fool or a liar, or that your memory is faulty. Derbyshire is wrong to discourage his readers—who will need a good portion of ambition to allow them to penetrate his book—in any way, and particularly to do so on false grounds.

And he is indeed asking a lot of his readers. In his 21st chapter he walks the reader through Riemann's explicit formula    where J(x) denotes    (a terminating series because  , which is the number of primes less than y, is zero when y < 2) and where the complex numbers  are the zeros of the zeta function in the critical strip. And I don't mean that he simply explains the definitions of all the terms. He also explains how the series   converges conditionally, so the order of the terms is of the essence and the convergence is very slow, and he actually provides numerical estimates of the various terms in the case  x = 20. Once he has completed this, having shown in detail and with clarity how the formula yields the known value  , he goes on to show the way in which Möbius inversion combines with this formula to give Riemann's explicit formula for  , taking for the sake  of illustration  the  case x = 1,000,000 and carrying it through very clearly to show how it really does work out to give .  (But he does not adequately explain how he evaluated the slowly converging "secondary terms." He gives them to five decimal places, but in his computation of J(20) he already confessed that 86,000 terms of the sum had to be computed to attain four-place accuracy for this term, and he certainly does not expect us to believe that he found the nine-place result he gives in that case by adding terms of the series!)

Can a beginner follow this chapter?  Not unless the beginner is very talented. To tell the truth, I had to read it pretty attentively. But it is interesting.  The talented beginner will learn from it, as I learned from it. And those who can't follow it are not being sold a bill of goods, not being encouraged to think they understand and appreciate something they don't understand at all, and not being condescended to. They can give it their best shot, and if they fail they can still admire it and still appreciate much of the rest of the book, and may someday come back to it when they are no longer beginners.

A parting thought. In my opinion, all three books grossly overstate the connection of RH to prime numbers. Derbyshire even chooses the title "Prime Obsession." True, an investigation of the distribution of primes and the Euler product formula led Riemann to RH, but Riemann himself quickly switched to another function he called  (it is the  value at  of , which, as Riemann proves, is an even function of t that is real on the real axis) and his actual hypothesis was that the zeros of  are real!  To me,  is a symmetrized version of —symmetrized to put the functional equation of   in a simple form and to put the interesting part of the function on the real axis—that is an entire function of one complex vari able. RH is simply the statement that its zeros are real. The connection with prime numbers may or may not play a role in explaining the amazing extent to which Riemann's hunch has been  borne out by massive modem computations unimaginable in his day. Despite this modem evidence in its favor, and despite its connection to a raft of fascinating and theoretically important "generalized Riemann hypotheses" that also stand up to computational scrutiny, no one seems to have any idea why it should be true. Who wants to be a millionaire?

Courant Institute of Mathematical Sciences
  New York University
New York 10012, USA
e-mail: edwards@cims.nyu.edu