American Scientist

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. John Derbyshire. xvi + 422 pp. Joseph Henry Press, 2003. $27.95.

The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics. viii + 340 pp. Karl Sabbagh. Farrar, Straus and Giroux, 2002. $25.

General interest in the strange world of mathematics and mathematicians has been much in evidence the past several years. In the early 1990s the solution of Fermat's Last Theorem caught the public's attention; several books about it subsequently appeared, and a short-lived off-Broadway musical was based on it. In 1998 A Beautiful Mind, Sylvia Nasar's gripping biography of the mathematician John Nash, became a popular success; last year the movie version won four Academy Awards. David Auburn's Proof enjoyed the longest run of any Broadway play of the past two decades, winning both a Tony Award and a Pulitzer Prize. The year 2000 was declared the year of mathematics, and several nations around the world issued stamps to commemorate the event.

Will the Riemann hypothesis be the next mathematical topic to captivate the general public? Publishers of the two books reviewed here appear to hope so. Prime Obsession and The Riemann Hypothesis present to the nonmathematician what is indeed the greatest unsolved problem in pure mathematics, describing its history, the men who have contributed to its understanding and their motivation for tackling it.

The roots of the problem go back to antiquity, when mathematicians started thinking about primes, the integers that are not a multiple of a smaller number greater than unity. They appear in Chinese manuscripts dating back to 500 B.C. and are discussed in Euclid's Elements, which he wrote around 300 B.C. Primes don't follow predictable patterns, and their distribution remains quite mysterious.

In 1737 the Swiss mathematician Leonhard Euler announced that he had found that the sum of the reciprocals of the squares of positive integers is Π²/6. His contemporary Daniel Bernoulli found this result sehr merkwürdig, very strange indeed. Euler went on, finding similar expressions for the sum of the reciprocals of even powers of the positive integers and studying the general properties of the sum of the powers of the positive integers, for an arbitrary exponent. This sum, considered as a function of the exponent s, is the Riemann zeta function ζ(s).

Two of Euler's discoveries stand out. One is a decomposition of the sum in question as a product of fairly simple factors associated with all primes, namely p^s/(p^s–1), where p is a prime and s the exponent we are dealing with. The other, conjectured by Euler and proved much later by Georg Friedrich Bernhard Riemann (1826–1866), was a hidden symmetry of this function, the so-called functional equation.

Primes and the zeta function remained separate for a while: The former were classified as belonging to arithmetic, and the latter was considered an object in analysis to be studied by calculus. Things changed overnight with the publication of Riemann's memoir, "On the Number of Prime Numbers Less Than a Given Quantity." Here Riemann solved the problem of how to study primes using the Euler product formula for the zeta function. He showed how to give a meaning to ζ(s) for every value of s, including complex numbers, and he found an exact formula for counting the number of primes up to a given quantity by means of a main term accounting for the statistical distribution of primes in the large, followed by smaller oscillatory correction terms determined by the solutions of the equation ζ(s) = 0. Here one must consider all solutions in complex numbers s = a + b√–1; for every such solution there is an associated correction term. These solutions are called the zeros of ζ(s). Riemann, on the basis of preliminary studies and numerical calculations, conjectured that for every complex zero, a = 1/2. This statement is the still-unproved Riemann hypothesis.

Initially the Riemann hypothesis was viewed just as a simple question in analysis, likely to be solved sooner rather than later. Time proved otherwise. The zeta function turned out to be the simplest case of a much larger class of objects, called L-functions, which should have properties similar to those of the zeta function. The values of L-functions at special points should have special significance, connected with deep facts of geometry. L-functions should admit a factorization entirely analogous to the product Euler found, a functional equation (the additional symmetry), and they should all satisfy a corresponding Riemann hypothesis. The consequences of proving this generalized Riemann hypothesis would extend well beyond information about the distribution of primes; thousands of results, forming a totally coherent set of statements, have been obtained by assuming these hypothetical properties of L-functions. But verification remains a major challenge, and the all-important Riemann hypothesis has not yet been proved (or for that matter, disproved) for any L-function. Any exception to it would create havoc in our understanding of arithmetic. This is why the Riemann hypothesis is so important and was one of the seven "Millennium Prize Problems" put forward by the Clay Mathematics Institute in May 2000 (http://www.claymath.org/Millennium_Prize_Problems/Riemann_Hypothesis).

John Derbyshire's Prime Obsession takes a historical tack. Part I is dedicated to the distribution of primes and the discovery of the fundamental role played by the zeta function, and it ends with an account of the proof of the Prime Number Theorem, which states that the density of primes is 1/log(n). Derbyshire begins with a riveting account of Riemann's times, his early life and his university years, up until his appointment at the age of 33 to the Berlin Academy and the publication of his famous memoir. This is followed by a discussion of Carl Friedrich Gauss and his views on the distribution of primes, together with a short but vivid biography of Euler. After this, other characters in the drama are introduced: Johann Peter Gustav Lejeune Dirichlet, Pafnuty Lvovich Chebyshev, Thomas Stieltjes (whose claim to have proved the Riemann hypothesis was never substantiated) and Jacques Hadamard, one of two men to prove independently the Prime Number Theorem.

Part II deals with the work of 20th-century mathematicians and with the zeta function itself, ending with a glimpse of the progress achieved in understanding it over the past 10 years. It presents a wider picture of the mathematical world, not limited to the Riemann hypothesis. Discussed first is David Hilbert, who selected the hypothesis as one of the 23 most important problems facing mathematicians in 1900. Next comes an account of the achievements of the English and German schools of analysis. G. H. Hardy, J. E. Littlewood and Edmund Landau are well presented, with a good description of the atmosphere of Cambridge and Oxford. We see the disintegration of German mathematics under Nazism. Derbyshire describes Carl Ludwig Siegel's effort in the early 1930s to decipher Riemann's unpublished notes, ending with Siegel's proof of the so-called Riemann-Siegel formula. This has become today the key to the numerical calculation of the zeta function. Work by Jørgen Pedersen Gram, Alan Turing and many others is described, culminating with the verification, in April 2002, that the first 50 billion solutions of the Riemann equation do in fact satisfy the Riemann hypothesis. Wow! Still, the hypothesis remains wide open.

The final chapters move at a quick pace and concentrate on ideas inspired by physics. Derbyshire rightly focuses on Hugh Montgomery's prediction (inspired by a conversation with Freeman Dyson) of a pair-correlation statistic for the zeros of the zeta function. (Contrary to what was once believed, the distribution of zeros is not completely random.) Derbyshire also describes Andrew Odlyzko's astonishingly accurate numerical verification of this pair-correlation conjecture and then moves quickly to Alain Connes's spectral analysis using adèles (a highly technical extension of the notion of number). Finally he speculates on the future, and the hope of finding at last a Hermitian operator whose resonances describe the zeros of the zeta function. This should stir the reader's imagination.

Interspersed with the historical narrative are tutorial chapters in which Derbyshire attempts to introduce nonmathematicians to the mathematics needed for a deeper understanding of the subject. Chapter 1 describes the trick of sliding a deck of 52 cards so as to make the deck overhang as much as possible without falling down; the purpose is to introduce readers to the harmonic series. Here Derbyshire has the air of a teacher desperate to attract the attention of bored students. The zeta function appears in chapter 5, where he takes nine pages to explain powers, all the way down to arithmetic for children. Many people will stop reading here, but that would be a mistake—the rest flows very well, and the remaining tutorial chapters, containing more meat, are more appetizing, notwithstanding an excessive use of diagrams. However, this material does interrupt the human story; it would have worked better as an appendix.

I like the historical account, which is well woven and accurate as far as I could check. A fair picture of the problem is presented, even though that picture is far from complete: The vast panorama of general L-functions dominating the landscape today is certainly beyond the scope of this work.

Altogether the author has succeeded in writing a very readable and interesting book. The appendix provides a funny song describing the Riemann hypothesis, written by Tom Apostol in 1955. It makes a fitting finish, showing that mathematicians also have a light side.

Karl Sabbagh's book, The Riemann Hypothesis, has a completely different structure. The material that forms its backbone was gleaned in a series of interviews with mathematicians who are experts on primes and zeta functions. (I was one of those interviewed, and the book includes a few brief paragraphs about me.) Sabbagh moves freely from serious information to trivia to anecdotal stories, sprinkling the text with small doses of mathematics to help the reader get a feeling for what is going on.

In chapter 1, the primes are presented as the building blocks of multiplication; when interviewed by Sabbagh, the mathematician Jon Keating compared them to pieces of Lego. Sabbagh explains Euclid's proof of an infinitude of primes; then he provides an old newspaper clipping announcing the discovery (made without the assistance of a computer) of a 72-digit prime number—which would be impressive except that, as he notes, the number is not prime. He also mentions the Great Internet Mersenne Prime Search (GIMPS), which makes use of the idle time on thousands of personal computers around the world and recently discovered a prime of more than 4 million digits. Offsetting this are Andrew Granville's comments that this is of almost no mathematical interest and a description of the searches by Harvey Dubner, a retired engineer, for unusual primes. The chapter concludes with excerpts from interviews with prominent mathematicians and with Gauss's observations on the distribution of primes.

The first five chapters ramble: A reference to Hilbert becomes the occasion to talk about Hilbert's 10th problem (the eighth is the Riemann hypothesis), and from there Sabbagh goes to Julia Robinson and her attack on the tenth problem, and then to its eventual solution by the young Russian mathematician Yuri Matijasevich. The collaboration between Hardy and Littlewood, who obtained important results about the zeta function, becomes the occasion to talk about the mathematical genius Srinivasa Ramanujan and to relate a number of amusing stories. (A fuller account of that collaboration and other stories can, by the way, be found in Littlewood's Miscellany, a gem of a book edited by Bela Bollobas.)

Then at the end of chapter 6 (an account of attempts to prove the Riemann hypothesis), we encounter the hero of the book, Louis de Branges—a good, but controversial, mathematician at Purdue University. His claim to fame is his celebrated solution in 1984 of the Bieberbach conjecture, a very old problem that had attracted a lot of attention from the specialists in the theory of complex functions. His controversial status stems from his numerous claims to have solved other famous problems (including the Riemann hypothesis) with proofs that have turned out to be fallacious in the end. In the view of most mathematicians, any major claim followed by dismal failure becomes a black mark that may take a long time to disappear.

So why has Sabbagh chosen de Branges as his hero, when experts regard his work with suspicion after his repeated attempts to "proof fix" the Riemann hypothesis? He candidly explains:

I chose de Branges for the simple reason that he told me he was putting the final touches to a proof. . . .

Here was a man who had actually been thinking about the Riemann Hypothesis for twenty years or more. Surely he’d know it as well as any other mathematician, even if he was barking up the wrong tree in his search for a proof. And there was always the Bieberbach Conjecture to his credit.

About half of the rest of the book is devoted to de Branges and half to presenting a picture of the world of mathematicians. On one side we find the Montgomery-Dyson statistics; Odlyzko's numerical work; Keating and Sir Michael Berry talking about the possible role of physics for understanding the problem; Connes's new ideas; an amusing account of a meeting in Oberwolfach in September 2001 about zeta functions; anecdotes; and more interviews. On the other side are entire chapters devoted to de Branges; these describe Sabbagh's meetings with him and convey an interesting picture of his personality. In the book's final chapter, Sabbagh quotes at length from the anonymous peer reviews of de Branges's 2002 grant application to the National Science Foundation, in which he asked for funding of his researches on the Riemann hypothesis. The application was denied.

The book ends with a group of short chapters called Toolkits, which explain logarithms and exponents, equations, infinite series, the Euler identity, graphs, and matrices and eigenvalues. These contain more detailed mathematics and are useful for a technical comprehension of the book. An appendix, "De Branges's Proof," is a translation of a presentation de Branges made (titled "The Riemann Hypothesis for Dirichlet Zeta Functions") at the Seminar in Number Theory at the Institut Henri Poincaré in Paris in May 2002. My own expert reading of this appendix has failed to reveal anything new; it is a rehash of de Branges's ideas of 1986, and it is a pity that Sabbagh did not get a serious technical review before putting it in his book. Sabbagh wants de Branges to solve the Riemann hypothesis. If de Branges succeeds, Sabbagh too will become famous. If de Branges fails, Sabbagh's book will go into oblivion.

This is an interesting book in many ways—lively, full of anecdotes and fun to read. The reader will find in it a picture not only of the Riemann hypothesis, but also of the strange world of mathematicians. In this respect Sabbagh's view is quite different from Derbyshire's historical approach, and the two books complement each other in many ways. However, Sabbagh's wish to entertain has resulted in an overall picture that is shallow and distorted, which greatly diminishes the value of the book.