Review of Prime Obsession

By Mark P. Silverman, Reviewer

Physicists 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 on 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 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 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 whose eigenvalues are the imaginary parts of the
zeta zeros.

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 10^371 digits! (For a physical
perspective on such a number, note there are approximately 10^80 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 Göttingen, "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.