In recent years there has been a raft of popular books on various topics in mathematics. Attention seems now to have turned to the Riemann Hypothesis; the book under review is one of at least three on that subject that have appeared this year. No doubt this sudden surge of interest in the Riemann Hypothesis can be attributed to the $1 million prize that Boston businessman Landon Clay has offered for its solution. Indeed, the first chapter of du Sautoy’s book is titled “Who wants to be a millionaire?” It seems unlikely that if, say, Paul Erdős had offered $100, it would have generated the same response.
At any rate, Marcus du Sautoy’s The Music of the Primes is an exposition of the Riemann Hypothesis for the notorious “general reader”. (On p. 77, he carefully explains that 1/2x is the reciprocal of 2x.)
But how is this possible? In order even to state the Riemann Hypothesis, we have to be able to mention the Riemann ζ-function. To say what that is, we must presume that the reader at least knows what the complex numbers are, and is able to think about functions of a complex variable. Then we have to talk about infinite series and the process of analytic continuation. Of course, it is not enough just to state the Hypothesis; the reason for the interest in it is its connection with the distribution of prime numbers. To see what that connection is, we have to bring in the Euler product for the ζ-series, which, of course, is based on the unique factorization property. All this just to be able to say what the Riemann Hypothesis is. If we want also to say something about the methods which have been used to try to prove the Hypothesis, we would have to bring in asymptotic approximations, eigenvalues of Hermitian operators, contour integrals, L-functions, and perhaps a great deal more.
It is not possible, of course, to make these things clear to the “general reader”, and du Sautoy doesn’t. His method, rather, is to write some words about these topics, with the intent, perhaps, of giving the reader the illusion of understanding. He begins by expatiating on the prime numbers. “Prime numbers are the very atoms of arithmetic” (p. 5). He tells us over and over again that they are just so, so important. “Their importance to mathematics comes from their power to build all other numbers” (p. 5). So I guess that without primes, we wouldn’t even have any other numbers.
Du Sautoy does indeed sketch (pp. 35–36) the proof that every positive integer can be factored into prime factors. Du Sautoy’s proof is the one which begins by considering a minimal counterexample. This seems an odd choice of argument, if the purpose is to reach the understanding of the non-mathematical reader. Indeed (p. 36), “when I tried this argument out on friends, they felt as if they had been cheated somewhere along the way.” Perhaps the point was simply to allow du Sautoy to use the catchy phrase ‘minimal criminal’. As far as I can tell, du Sautoy never mentions the fact that the prime factorization is unique. Possibly he felt that the concept of uniqueness is too subtle; or maybe the “general reader” would think that “unique” just means “cute”. (But on p. 292 du Sautoy discusses what makes mathematics “so unique” among the sciences, so for him, apparently, uniqueness is a matter of degree.)
Du Sautoy then introduces the prime-counting function π(N), and discusses Gauss’s estimate that π(N) is asymptotic to N/log(N). Of course, this requires an explanation of the natural logarithm. Here it is (p. 49): “e turns out to be as important in mathematics as the number π, and occurs all over the mathematical world. This is why logarithms to the base e are called ‘natural’ logarithms.”
In the third chapter, we encounter the imaginary numbers. “The square root of minus one, the building block of imaginary numbers, seems to be a contradiction in terms. Some say that admitting the possibility of such a number is what separates the mathematicians from the rest. A creative leap is required to gain access to this bit of the mathematical world” (p. 66). I don’t see that du Sautoy ever explains why the square root of –1 is not a “contradiction in terms”. It’s just that “nineteenth-century mathematicians were brave enough to believe in new modes of thought which challenged the accepted ideas of what constituted the mathematical canon” (p. 69). If the reader is beginning to get a little confused here, the reason is clear: “Some say that admitting the possibility of such a number is what separates the mathematicians from the rest.” (It is really shameful, I think, to perpetuate this mysticism about imaginary numbers. I tell my students that complex numbers are just two-dimensional vectors which can be multiplied. No mystery.)
We then learn that Dirichlet, in his proof of the theorem on primes in arithmetic progressions, had made use of Euler’s ζ-function:
ζ(x) = 1–x + 2–x + 3–x + … n–x + …
What is the connection with primes? Well, Euler had found the factorization
ζ(x) = (1 + 2–x + 4–x + …)(1 + 3–x + 9–x + …)…(1 + p–x + (p2)–x …) …
This seems to be related to prime factorization of integers, since du Sautoy explains (p. 80): “in one equation was encapsulated the fact that every number can be built by multiplying together prime numbers”. But du Sautoy never shows how Euler’s factorization of the ζ-series is connected with prime factorization of integers. I guess that that would require some simple algebra. So we are left with du Sautoy’s assertion that primes are important, and with his assertion that they are somehow connected with the ζ-function.
Apparently (p. 72), it was Euler who first used imaginary exponents. “To his surprise, out came waves which corresponded to a particular musical note.” So now we know what imaginary exponents mean.
The next step is Riemann’s study of the ζ-function as a function of a complex variable. (Du Sautoy doesn’t mention that the same idea was part of Dirichlet’s proof.) Du Sautoy describes Riemann’s work in terms of a certain surface lying above the complex plane. It’s not the graph of the ζ-function, of course; that graph lies in 4-dimensional space (p. 85). To make a 3-dimensional graph, “we have a number of choices for what to record as the height of the landscape above each imaginary number in the map on the table top. There is, however, one choice of shadow which retains enough information to allow us to understand Riemann’s revelation.” Du Sautoy never mentions that the resulting “landscape” is the graph of the absolute value of the ζ-function.
Now, in order to state Riemann’s hypothesis, we have to extend the original Dirichlet series for the ζ-function by analytic continuation. “Riemann succeeded in finding another formula that could be used to build the missing landscape to the west” (p. 88). It turns out that the zeros of the ζ-function are crucial. “Any mathematical cartographer who knew how to plot on the two-dimensional imaginary map the points where the landscape fell to sea level could reconstruct everything about the entire landscape” (p. 88). (Obviously, this is false, in general; just multiply by an exponential. I suppose du Sautoy is thinking of the Hadamard product representation — see Edwards, Riemann’s Zeta Function, chapter 2.) And finally we have Riemann’s conjecture that these zeros lie on the line Re(s) = 1/2. “Riemann had finally found the mysterious pattern that centuries of mathematicians had been yearning to see as they stared at the primes” (p. 99).
As there is clearly no possibility that the reader will be able to understand all this, du Sautoy compensates by tarting up his exposition with purple prose. Every advance is a “breakthrough”. Every discovery is “stunning”. Every new idea is “revolutionary”. Thus, Gauss’s modular arithmetic (which du Sautoy refers to as “Gauss’s clock calculator”) “revolutionised mathematics at the turn of the nineteenth century” (p. 21). Riemann’s Hypothesis, naturally, is the “Holy Grail” (p. 2). (Mathematics apparently differs from Arthurian legend in having several “Holy Grails” [p. 187]; for example, Riemann’s exact formula for π(N) [p. 91], the Prime Number Theorem [p. 103], and a polynomial which generates all primes [p. 200].) More generally, du Sautoy describes mathematics and mathematical work in religious terms: “revelation”, “epiphany”, “article of faith”, “anathema” and “heresy”. At the same time, he continually invokes magic: “magic formula”, magic of logarithms, magic of primes, “Riemann’s magic ley line” (the critical line for the ζ-function), Erdős as “wizard”. (In our magic-obsessed society, I suppose few readers will feel any incompatibility between mathematics-as-religion and mathematics-as-magic.)
As du Sautoy tells the story of the mathematicians since Riemann who have worked on the Riemann Hypothesis, he seems to feel that no old chestnut can be left untold, no matter how remote from his subject. So we hear about Euler vanquishing Diderot and proving the existence of God by algebra (pp. 42–43), Hardy trying to trick God by wearing four sweaters to a cricket match (p. 121), Gödel’s fear of being poisoned (p. 179), and Nevanlinna saving André Weil from being executed as a spy (p. 294).
Du Sautoy has somehow been able to learn the innermost thoughts of his characters. “A strange new vista began to open up before Riemann’s eyes. The more he scribbled away on the pages which covered his desk, the more excited he became” (p. 82). “Hilbert found something even more unsettling: no one had actually proved that the theory of numbers itself did not contain contradictions. Suddenly Hilbert was reeling” (p. 111). “So Gauss formed a picture in his mind of how Nature might have decided which numbers were going to be prime and which were not. Since their distribution looked so random, might tossing a coin not be a good model for choosing primes? Did Nature toss a coin — heads it’s prime, tails it’s not? Now, thought Gauss, the coin could be weighted so that instead of landing heads half the time, it lands heads with probability 1/log(N)” (p. 55). According to du Sautoy, these ideas are explained in a letter Gauss wrote to the astronomer J. F. Encke, on Christmas Eve, 1849. They are not. Gauss says nothing about flipping coins, nor does he interpret the distribution of primes probabilistically. (An English translation of this letter can be found in Larry Goldstein’s 1973 Monthly article on the history of the Prime Number Theorem.) But once having made up this story, du Sautoy treats it as a simple fact: “Gauss had used the idea of tossing a prime number coin to guess at the number of primes” (p. 165).
According to du Sautoy (p. 41), “The young Euler’s precocious mathematical talents, however, had brought him to the notice of the powers that be. Euler was soon being courted by the academies throughout Europe.” Actually, Euler depended on the influence of his friends Nicholas and Daniel Bernoulli to obtain a position at the St. Petersburg Academy — in physiology! Although Euler had placed second in the Paris Academy competition of 1727 (with a paper on the masting of ships), he first attained mathematical fame by his solution in 1735, six years after his arrival in St. Petersburg, of the “Basel problem” — the evaluation of ζ(2) = π2/6.
Du Sautoy’s prose is larded with clichés. For Ramanujan, dissecting frogs and rabbits would have been “beyond the pale” (p. 135). Gödel’s Incompleteness Theorem was a “major body blow” to mathematicians (p. 181). Also, it “struck at the heart” of Hilbert’s “world-view” (p. 178). It was Riemann who would “truly unleash the full force” of the “hidden harmonies that lay behind the cacophony of the primes” (p. 58). The work of Diffie and Hellman “heralded a new era” in encryption (p. 227). “This is why the likes of Euler and Cauchy were so against the graphical depiction of imaginary numbers” (p. 113). Aside from Euler and Cauchy themselves, who might their “likes” have been?
Du Sautoy several times writes of “navigating” Riemann’s imaginary “landscape” (for example, p. 106). There could be no clearer indication, I think, that he pays no attention to the words he is using. (And on p. 287, scientists following Riemann were “increasingly less” interested in the “crossover” between mathematics and physics.) On p. 237, du Sautoy uses the phrase “begs the question” to mean “raises the question”. Are there no longer any editors?
I assume that the publishers, Harper Collins, know their business, and that consequently there must be a market for this kind of book. Some readers may find du Sautoy’s account entertaining, in which case I suppose that there are less productive ways of spending their time. From another point of view, however, we might ask whether such a book is helpful to the mathematical community.
It is possible, I suppose, that politicians who read the book might be inspired to vote lots of money for mathematical research. Or deans who read it might treat the mathematics department more favorably. These possibilities are perhaps not as far-fetched as they may seem. I have known colleagues from other disciplines who have become very enthusiastic about certain popular expositions of mathematical ideas.
But will perhaps some talented young person be inspired by the book to become a mathematician? Also not impossible; many of us, I suppose, were fascinated by E. T. Bell’s Men of Mathematics, which is pretty free with the truth. However, for all his racy stories, Bell was not as condescending as du Sautoy. Bell actually did try to explain mathematical ideas —look at his discussion of the derivative, pp. 98–99, which is better than what you will find in most calculus textbooks. In any case, the aspiring mathematician may find du Sautoy’s admonition that “only those with a special aesthetic sensibility are equipped to make mathematical discoveries” (p. 78) rather discouraging.
For a student who has adequate preparation and wants to learn about the Riemann Hypothesis, I can think of no better place to begin than Harold Edwards’s fine book on the ζ-function, now readily available in a cheap reprint from Dover. Edwards includes an English translation of Riemann’s original paper. An introduction to more recent work is given in Brian Conrey’s article in the March 2003 Notices of the AMS.
- H. M. Edwards, Riemann’s Zeta Function, Academic Press, 1984; Dover reprint, 2001.
- E. T. Bell, Men of Mathematics, Simon & Schuster, 1937.
- L. J. Goldstein, “A History of the Prime Number Theorem”, American Mathematical Monthly, 80, 1973, pp. 599–615.
- J. Brian Conrey, “The Riemann Hypothesis”, Notices of the American Mathematical Society, 50, 2003, pp. 341–353. Available online.
For Euler’s early career, see
- Ronald Calinger, “Leonhard Euler: The First St. Petersburg Years (1727–1741)”, Historia Mathematica, 23, 1996, pp. 121–166.
- The article “The Spectrum of Riemannium”, by Brian Hayes, American Scientist, 91, 2003, pp. 296–300, discusses some of the material in du Sautoy’s chapter 11 with less gossip, but greater clarity. Available online.
Other books on the Riemann Hypothesis:
Stacy G. Langton (firstname.lastname@example.org) is Professor of Mathematics and Computer Science at the University of San Diego. He is particularly interested in the works of Leonhard Euler, a few of which he has translated into English.