Joseph
Henry Press, 412 pages, $24.95
reviewer James Franklin
With the proof
of Fermat's Last Theorem in 1994, Derbyshire says, "the Riemann
Hypothesis is now the great white whale of mathematical research".
Even before that, it was regarded by mathematicians as the more
significant problem - though not as old as FLT, it is more central
to mathematics and probably a good deal harder.
And harder to
explain. Of the two new books offering an account for a popular
audience, Prime Obsession and Karl Sabbagh's The Riemann Hypothesis
(Farrar, Straus and Giroux, $25), Derbyshire's offers the better
insight. Sabbagh's is a well-written book with interesting stories
from mathematicians working in the field, but Derbyshire is a talented
expositor determined to make the reader understand some serious
mathematics. A general reader with some memory of high school algebra
who is willing to concentrate will come away with a grasp of what
the problem is and why insiders are excited. Mathematicians in other
fields will deepen any superficial understanding they may have,
as well as picking up some new ideas on how to explain mathematical
ideas.
The importance
of the Riemann Hypothesis comes from its close connection to one
of the most basic phenomena connected with numbers, the distribution
of primes. At first glance, numbers all look much the same except
for their size. They are not. Twelve eggs can be arranged as a rectangle
of 6 eggs by 2, or 3 by 4. That cannot be done for 11 or 13 eggs.
11 and 13 are primes, numbers that cannot be written in any way
as a product of smaller numbers. The primes less than 50 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Between 980
and 1000, the only primes are 983, 991 and 997. Between 9980 and
10000, there are none. It can be seen that the primes thin out as
we go further along the numbers (though they never run out). There
is, however, an irregularity or jaggedness to the way they thin
out. It is believed, though not proved, that there are an infinite
numbers of prime pairs - that is, however far out we go, there is
always an occasional pair of odd numbers only two apart that are
both primes (like 41 and 43). On the other hand, there are indefinitely
long stretches of numbers with no primes at all. It is easy to understand
why: to get a stretch of, say, 100 numbers without a prime, multiply
together all the numbers from 1 to 101 and call the answer x. Then
the 100 numbers x + 2, x + 3, x + 4, … , x + 101 are all non-primes
(since x + 2 is divisible by 2 because x and 2 both are, x + 3 is
divisible by 3, and so on). Thus the sequence of prime numbers,
though it is a matter of absolute necessity and the same in all
possible worlds, has the interplay of overall orderliness with local
irregularity that we are accustomed to in sequences of throws of
dice and coins. Einstein may or may not have been right to say that
God does not play dice with the universe, but He certainly does
not play dice with the primes. Numbers are not subject to chance
or to any will, human or divine. That is what gives questions about
the distribution of primes their peculiar fascination.
The local irregularities make it hard to answer the "big-picture"
question: at what rate do the primes thin out? What is the average
density of primes - if we take a block of 1000 consecutive numbers
around, say, 1 trillion (1,000,000,000,000), what proportion of
them should we expect to be primes? The answer is given by the celebrated
Prime Number Theorem, whose proof was one of the glories of late
nineteenth century mathematics. The density of primes near the number
N is about 0.434/log(N) (where log(N), called the logarithm of N,
is the number of digits in N). The important item in this formula
is not the 0.434, which relates to the fact that we have decided
to write numbers in base 10 (if we wrote them in base 2 as computers
do, the number of digits in N would be greater). The significant
thing is that the density of primes thins out logarithmically -
since 1 trillion has twice as many digits as 1 million, the density
of primes around 1 trillion is half what it is around 1 million.
Riemann's 1859
paper, in which he introduced his Hypothesis, is a bold series of
moves which gives a formula not only for the average density of
primes but for all the irregularities as well. Late in his book,
Derbyshire ambitiously but successfully unpacks this short and difficult
paper, and explains how Riemann gives an exact formula for the deviations
from the average density, so that one can calculate exactly how
many primes there are in any block of numbers. The formula has a
downside, however. It expresses the answer in terms of some mysterious
entities called the zeros of the Riemann zeta function. There are
an infinite number of these beings, and the Riemann Hypothesis says
of them, "All the non-trivial zeros of the Riemann zeta function
have real part one half." Explaining from a standing start
what the Riemann zeta function and its zeros are in only half a
book is not easy, and Derbyshire proves himself a leading mathematical
communicator in being able to do it. "If you don't understand
the Hypothesis after reading my book", he says, "you can
be pretty sure you will never understand it." He is right.
The book is
not all tough mathematics. Included, for example, is the bizarre
connection between the way the zeros of the zeta function occur
and the way some quantum mechanical systems are spaced. There is
something on the use of prime numbers in internet security. There
is some history, including the little there is to know about Riemann
himself. He was pious, shy, depressed, and died of tuberculosis
aged not quite forty. On the real world, his impact was minimal.
When he went through the door into his study and tapped into the
abstract world, he made enormous advances in several different mathematical
fields. "Riemann's mathematics has the fearless sweep and energy
of one of Napoleon's campaigns."
Derbyshire handles with kid gloves, as well he might, a question
unavoidable when talking about an unproved hypothesis, that of probabilistic
reasoning in pure mathematics. He writes that "Everybody knows
that in mathematics you must prove every result by strict logic."
That is true in the sense that a strict proof of everything is sought,
but it is not true if it means that anything not proved is not yet
part of mathematics. If that were true, there would be no book about
the Riemann Hypothesis, since it is not proved. So is the evidence
for its truth good? Should we gather evidence for and against it,
as if it were a defendant in a court of law? Since the Hypothesis
has the same logical form as "All swans are white", the
most direct sort of evidence comes from calculating the zeros and
checking if their real part is indeed a half. The zeros are ordered,
so one can speak of the first one, second one, and so on. It was
shown in 1903 that the first 15 zeros do have real part a half,
and both people and machines have been busy since. 50 billion is
announced, but it is hard to keep track, since there is a co-operative
project using spare computer time that claims to be knocking over
a billion zeros a day. It is one of the largest inductions in history.
In such abstract areas, however, it is not surprising that there
are more subtle reasons bearing on the question. Most experts are
firmly convinced that the Hypothesis is true, but there still are
a few sceptics. There is just a little reason to think that though
there are no small counterexamples (zeros with real part not a half),
there could be some very large ones, ones far beyond the reach of
any feasible calculation. There is something to be said for the
opinion of the mathematician George Polya, that pure mathematics
is the best place to appreciate probabilistic reasoning. For in
mathematics, there are no distractions from subjective factors or
laws of nature. The hypothesis confronts the evidence in pure logical
space.
Will the Riemann
Hypothesis be proved soon? Derbyshire takes the risk of making a
fool of himself and puts his prediction on the line: no. The ideal
reader of his book, then, is an obsessive fifteen-year-old genius
like the young Gauss, who often spent an "idle quarter of an
hour" tallying the primes in blocks of a thousand. The problem
will most likely still be there when that reader is old enough to
tackle it.
