Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

by John Derbyshire
Plume/Penguin, 2004, New York

Reviewed in American Journal of Physics by Mark P. Silverman

Physicists ordinarily encounter numerous special functions in their professional careers, and it may seem at first blush that the Riemann zeta function is just another creature in a mathematical zoo of endless variety. Or so it seemed to me when, in the 1980s, I first encountered the zeta function while deriving the quantum spatial and temporal coherence functions of thermal electrons. Were someone to have told me then that he intended to write an epic saga about the zeta function, which would keep me, a physicist, not a mathematician, glued to my seat until I finished reading it, I would have laughed in disbelief. Nevertheless, this is precisely what author John Derbyshire has accomplished.

Presented in a conversational style, but with the meticulous attention to detail of a well-composed detective novel, Prime Obsession tells of the origin, evolution, and significance of a mathematical conjecture with deep ramifications throughout many fields of mathematics and surprising physical implications still to be explored fully. Seamlessly the author weaves together the "world lines" of Riemann and the eminent mathematicians who either motivated or followed up his work, explaining carefully and readably the essential mathematical contributions made by each.

Like quarks to a physicist or the chemical elements to a chemist, prime numbers form an irreducible set of fundamental entities to a mathematician. Although it has been known for more than two millennia that the number of primes is unlimited, the question of how densely primes are distributed along the number line was not answered definitively until just shortly before the start of the 20th century. This result is known as The Prime Number Theorem (...not merely a prime number theorem, as Derbyshire points out in reflecting upon its significance). Although prime numbers belong to arithmetic and the zeta function to analysis, there is an intimate and fascinating connection between the two through the nontrivial zeros of the zeta function. This connection, which lies at the core of Prime Obsession, finds its most far-reaching, and as yet unproved, expression in the Riemann Hypothesis (RH), a conjecture whose demonstration, according to Derbyshire, has become the foremost outstanding problem in mathematics today.

The conjecture, while evidently very difficult to prove (since mathematicians are still working at it), is trivially easy to state. However, it would be pointless to do that here, for without the rich background provided in the author's narrative, the RH would only sound like one of those highly specialized, "practically useless" mathematical theorems over which physicists roll their eyes upward and wonder why anyone would care. The RH, however, is well worth a physicist's wonder. In the chapter "Number Theory Meets Quantum Mechanics", the author discusses the statistical properties of the zeta zeros and their strange similarities to the statistical properties of the eigenvalues of Gaussian-random Hermitian matrices, which arise in the dynamical analyses of multiparticle systems like heavy atomic nuclei. "What on earth," the author asks rhetorically, "does the distribution of prime numbers have to do with the behavior of subatomic particles?" Some day, perhaps, a physicist may answer that question.

The mysterious connection with physics, however, goes even further. In a subsequent chapter "The Riemann Operator and Other Approaches," Derbyshire discusses the possibility that the nontrivial zeros of the zeta function might actually be the eigenvalues of some physically significant "Riemann operator". Coincidentally (although I did not realize this until I read Derbyshire's book), in the same year that I reported the coherence functions of chaotic electron states in terms of zeta functions, Michael Berry published a paper on the zeta function as a model for quantum chaos. Berry argued that the Riemann operator, if it exists, represents a semiclassical chaotic system, whereby the imaginary parts of the zeta zeros are the eigenvalues of this system.

Although the RH has not been proven, computational work has established that it is true for about 100 billion zeros of the zeta function. Is it conceivable that with such overwhelming numerical support, the conjecture could ever be wrong? As a physicist, one of the more sobering lessons I drew from Prime Obsession concerns the potential fragility of inductive reasoning.

Scientists must reason inductively; the laws of physics, after all, are not mathematical theorems, but ultimately derived from and tested by experience. Nevertheless, if you knew the Sun rose every morning for the past 4.5 billion years - i.e. about 2 million million mornings - would you be confident the Sun would rise again tomorrow? Probably. Why? Because most of us - physicists included - find it hard to imagine something that could produce an expected result so often and then somehow fail to produce it. But mathematicians like Derbyshire can imagine these things. In his narrative on the prime number theorem, the author discusses two functions (the prime number counting function and the log integral) which for all practical (i.e. calculable) purposes look like they will never cross, but can be shown in fact to cross an infinite number of times starting at a number beyond any that present or future computers can handle, a number with about 10371 digits! (For a physical perspective on such a number, note there are approximately 1080 protons in the visible universe.)

Apart from the pleasure of seeing so many interesting connections being made before my eyes, I especially enjoyed reading the author's sympathetic account of Riemann's life. Like Maxwell's, Riemann's life was tragically short. But like Faraday's, Riemann's name is associated with a host of diverse achievements: besides Riemann zeta function and Riemann hypothesis, there are Riemann integral, Riemannian geometry, Riemann surface, Riemann curvature tensor, and Riemann coordinates, to cite but a few. It is impossible to think about general relativity without thinking of Riemann's name - and not solely because of the non-Euclidian geometry he created. I recommend to every physicist to read Riemann's Habilitation lecture at Goettingen, "On the hypotheses that lie at the foundations of geometry", which anticipates by more than a half century Einstein's geometric approach to understanding gravity.

Derbyshire has written an absorbing account of an extraordinary mathematician whose epochal works, even in the "purest" mathematical realms, illuminate the conceptual recesses of the physical world.

About the author

Mark P. Silverman is Jarvis Professor of Physics at Trinity College. He wrote of his investigations of light, electrons, nuclei, and atoms in his books Waves and Grains: Reflections on Light and Learning (Princeton, 1998), Probing the Atom (Princeton, 2000), and A Universe of Atoms, An Atom in the Universe (Springer, 2002). His latest book Quantum Superposition (Springer, 2008) elucidates principles underlying the strange, counterintuitive behaviour of quantum systems.