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| The
Indivisible Man By Enrico
Bombieri 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 pi-squared / 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 Zeta(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 Zeta(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
Zeta(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 Zeta(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. 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. |
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