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),
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 is
an infinite number 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).
There are indefinitely long stretches of numbers, however, 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. Fifty billion is announced, but it is hard
to keep
track, since there is a cooperative 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 skeptics. 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 Pólya, 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.