The April 30, 1945 issue of Time contained an article about Hans Rademacher's failed attempt to disprove the Riemann Hypothesis (RH). According to the article, "No layman can understand it [the Riemann Hypothesis], and no mathematician has ever proved it." Fifty-eight years later, the Riemann Hypothesis remains open, but John Derbyshire (pronounced "DAH-bi-shuh") has taken on the task of explaining it to the layman. He brings solid credentials to the task, both in mathematics and in writing. His educational background is in mathematics ("I hit the wall with a topic called Functional Analysis"), he regularly writes a column for National Review and he is the author of the novel Seeing Calvin Coolidge in a Dream. In the preface to this book, he writes "My original goal was, in fact, to explain the Riemann Hypothesis without using any calculus at all. [Emphasis in the original.] This proved to be a tad over-optimistic, and there is a very small quantity of very elementary calculus in just three chapters, explained as it goes along."
For a reader with more sophistication than Derbyshire's intended reader — say someone who has had a course in complex analysis — we can describe RH as follows. Take s to be a complex variable, and define the Riemann zeta-function as
This sum converges when the real part of s is greater than 1, and we can extend this definition via analytic continuation. The easiest way to do this is to note that
The sum on the right-hand side converges when Re(s) > 0; in other words, this equation gives an analytic continuation of ζ(s) to the half-plane Re(s) > 0. Given that background, it is now possible to state the Riemann Hypothesis: If ζ(ρ) = 0 and ρ = β + iγ, where β > 0, then β = 1/2. In other words, all zeros of ζ(s) in the half-plane Re(s) > 0 have real part 1/2.
Stating the Riemann Hypothesis this way is like describing playing the violin as dragging horsehair over catgut strings — the description is technically correct, but it fails to explain why so many people find the subject so beautiful and so fascinating. The attraction of the Riemann zeta-function lies in its connection to the primes. This connection comes through the Euler product formula, or as Derbyshire refers to it, "The Golden Key." This states that when the real part of s is greater than 1,
where the product runs over all primes p. For the reader who has never seen a proof of this equation (or even one who has), I recommend Euler's original proof as presented in chapter 7 of this book.
Here is one simple illustration of the power of the Euler product formula. From our original series definition or from the alternating series version, one sees that ζ(s) has a pole at s = 1. A finite product is everywhere convergent, so we have another proof that there are infinitely many primes. Moreover, by using the power series expansion of log(x) around x = 1, one can prove a stronger statement; namely, that the sum of the reciprocals of the primes is a divergent series.
More analytic information about the zeta-function yields more information about the primes. A good example — one that originally motivated Riemann — is the Prime Number Theorem (PNT). Let π(x) be the number of primes up to x, and let Li(x) be the integral from 2 to x of dt/log(t). PNT states that π(x) is asymptotic to Li(x) as x goes to infinity. It turns out that PNT is equivalent to showing that ζ(s) has no zeros on the line Re(s) = 1. This was proved by Hadamard and de la Vallée Poussin (independently) in 1896. A few years later, de la Vallée Poussin proved that there is a "zero-free" region to the left of Re(s) = 1, and this gave the prime number theorem with an error term. If we could prove the Riemann Hypothesis, then we could prove that
for any ε > 0. Conversely, if we could prove that this estimate is true for all ε > 0, then the Riemann Hypothesis would follow.
Derbyshire covers all of the above and more, but of course, he moves at a slower pace and assumes the reader has much less mathematical sophistication. He also tells the history of the mathematicians who have contributed to our understanding of the zeta-function. His layout is unusual; he uses the odd-numbered chapters for the mathematical story and the even-numbered chapters for the history. This approach works very well — the difficult technical pieces are interlaced with an entertaining and well-told historical narrative. A less technically inclined reader might concentrate on the history and lightly skim the odd-numbered chapters. On the other hand, the mathematically inclined should not skip the even-numbered chapters — they tell a good story that should not be missed. Derbyshire gives sketches of the many people who have contributed to our understanding of the zeta-function — Euler, Gauss, Dirichlet, Landau, Hardy, Littlewood, and others. Riemann is, of course, the central focus of the book. We know relatively little about Riemann's personality; the principal source is a short memoir written by Dedekind, ten years after Riemann's death. He was an extremely shy man whose only ties were with his family and other mathematicians. His health was never good, and he died at the age of 40. He wrote only one paper in the theory of numbers — "Über die Anzahl der Primzahlen unter einer gegebenen Grösse," which translates to "On the Number of Prime Numbers Less Than a Given Quantity." As Derbyshire writes, "Mathematics has not been the same since."
The mathematical treatment is leisurely at the beginning. At times, the author underestimates the mathematical sophistication of his likely readers; for example, there is a detailed discussion of the laws of exponents. At other times, he overestimates his readers. After describing the Euler product and its implications, the author discusses Riemann's explicit formula for π(x) in terms of the complex zeros ρ = β + iγ of ζ(s). This is important material, but it will be rough going for most readers. Mathematically expert readers may prefer von Mangoldt's simpler explicit formula, which Derbyshire mentions in passing. Accounts of this can be found in Davenport, chapter 17 or Edwards, chapter 3. Overall, the author does a good job with the mathematical explanations. There are a few errors, but nothing egregious. Moreover, Derbyshire has a list of errata and comments on his web page http://olimu.com/Riemann/Errata.htm.
The literature on the zeta-function is vast, and a book like this has to be very selective. However, Derbyshire does manage to mention some of the most active strands of current work; in particular, he discusses the pair correlation of zeros and the connection with eigenvalues of random Hermitian matrices. The study of pair correlation was initiated by Hugh Montgomery in the early 1970s, and he learned of the connection to eigenvalues of random Hermitian matrices in conversation with Freeman Dyson at tea at the Institute for Advanced Study. This conversation has become rather famous; Deift calls it "one of the most celebrated denouements in mathematics in recent years." As with any well-known and oft-repeated story, highly embellished versions exist; see, for example, the "screenplay version" of Hayes. Derbyshire goes to the original source and tells the story in a single long quote; he informs me (email communication) that he got the quote by transcribing videotapes of a lecture that Montgomery gave at the AIM Seattle Conference in 1996. To help spread the most reliable version of the story, I give an excerpt here.
I [Montgomery] took afternoon tea that day in Fuld Hall with [Sarvadaman] Chowla. Freeman Dyson was standing across the room. I had spent the previous year at the Institute and I knew him perfectly well by sight, but I had never spoken to him. Chowla said: "Have you met Dyson?" I said no, I hadn't. He said: "I'll introduce you." I said no, I didn't feel that I had to meet Dyson. Chowla insisted, and so I was dragged reluctantly across the room to meet Dyson. He was very polite, and asked me what I was working on. I told him I was working on the differences between the non-trivial zeros of Riemann's zeta-function, and that I had developed a conjecture that the distribution function for those differences had integrand 1 - (sin(πu)/πu)2. He got very excited. He said: "That's the form factor for the pair correlation of eigenvalues of random Hermitian matrices!"
Who should read this book? I recommend it for advanced undergraduate mathematics majors, especially those planning graduate study. They will learn some good mathematics presented in an interesting way, and they will see some good writing — something graduate students should see as much as possible. Students going into analytic number theory should follow up with detailed technical surveys such as Titchmarsh and Ivic.
What about those with less mathematical sophistication? As a test, I asked my son Matt to read the book. Matt is as a computer programmer, and his last math course was differential equations, which he took seven years ago. He was able to read the whole book, and, although he is not ready to start research in analytic number theory, he did gain some appreciation of the Riemann Hypothesis and of the Euler product formula. His favorite parts of the book were the mathematical anecdotes, particularly the one about Hilbert's graveside oration (pages 186-187). Readers with less mathematical sophistication — those with minimal or no calculus — will find the learning curve very steep. However, I agree with Derbyshire's statement that "if you don't understand the Hypothesis after finishing my book, you can be sure that you will never understand it."
Professional mathematicians are, of course, big consumers of popular mathematical books, and those with specialties outside of number theory should get a good feel for the zeta-function from this book. They can follow up with a more technical discussion of recent developments in Conrey's article in the Notices of the AMS. And those who know and love ζ(s) will see a familiar story entertainingly told, and they will also see a few new sides of our multifaceted and mysterious friend.
- J. B. Conrey, "The Riemann Hypothesis", Notices of the American Mathematical Society 50 (2003), 341–353. Available online.
- J. Derbyshire, "Joining the delights of reading and the fascinations of math", Boston Globe, September 19, 1999.
- H. Davenport, Multiplicative Number Theory, third edition, revised and with a preface by H. L. Montgomery, Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000.
- P. Deift, "Integrable Systems and Combinatorial Theory", Notices of the American Mathematical Society 47 (2000) 631–640. Available online.
- H.M. Edwards, Riemann's Zeta Function, Academic Press, New York, 1974. Republished by Dover Publications, Mineola-New York, 2001.
- B. Hayes, "The Spectrum of Riemannium", American Scientist 91 (2003) 296-300. Available online.
- A. Ivic, The Riemann Zeta-Function, John Wiley & Sons, 1985; reprinted Dover 2003.
- E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, second edition, edited and with a preface by D. R. Heath-Brown, The Clarendon Press, Oxford University Press, Oxford, 1986.
S.W. Graham is currently a Professor of Mathematics at Central Michigan University. He can be reached at firstname.lastname@example.org.