Fame and fortune await the person who cracks the
greatest problem in mathematics. And that could be any
day now, says Erica Klarreich
WHEN G. H. Hardy faced a stormy sea passage from
Scandinavia to England, he took out an unusual insurance
policy. Hardy scribbled a postcard to a friend with the words:
"Have proved the Riemann hypothesis". God, Hardy reasoned,
would not let him die in a shipwreck, because he would then
be feted for solving the most famous problem in mathematics.
He survived the trip.
Almost a century later, the Riemann hypothesis is still
unsolved. Its glamour is unequalled because it holds the key to
the primes, those mysterious numbers that underpin so much
of math- ematics. And now whoever cracks it will find not only
glory in posterity, but a tidy reward in this life: a $1 million
prize announced this April by the Clay Mathematics Institute in
There are signs that the great prize might soon be claimed,
and the most promising approaches come not from pure
mathematics, but from physics. Researchers have discovered a
deep connection between the Riemann hypothesis and the
physical world--a connection that could not only prove the
hypothesis, but also tell us something profound about the
behaviour of atoms, molecules and even concert halls. One
mathematician has followed this lead into a very strange
place, seeking a solution in an intricately twisted space with
infinitely many dimensions.
Yet the primes seem simple enough at first glance. They are
those numbers, like 2, 3, 5 and 7, that are only divisible by 1
and themselves, although 1 isn't included among them. Primes
are the atoms of the number system, because every other
number can be built by multiplying primes together.
Unfortunately there is no periodic table for the primes--they
are maddeningly unpredictable, and finding new primes is
mostly a matter of trial and error.
In the 19th century, mathematicians
found a little order in this apparent chaos.
Even though individual primes pop up
unexpectedly, their distribution follows a
trend. It's like tossing a coin. The result is
unpredictable, but after many coin tosses
we expect roughly half heads and half
tails. The primes get rarer as you look at
larger and larger numbers (see Diagram), and mathematicians
found that this thinning out is predictable. Below a given
number x, the proportion of primes is about 1/ln(x), where
ln(x) is the natural logarithm of x. So, for example, about 4 per
cent of numbers smaller than ten billion are prime.
So far so good. But that "about" is very vague. Numbers are
products of pure logic, and so surely they ought to behave in
a precise, regular way. Mathematicians would at least like to
know how far the prime numbers stray from the distribution.
Georg Riemann found a vital clue. In 1859, he discovered that
the secrets of the primes are locked inside something called
the zeta function. The zeta function is simply a particular way
of turning one number into another number, like the function
"multiply by 5". Riemann decided to see what would happen if
he fed the zeta function complex numbers--numbers made
from a real part (an ordinary number) and a so-called
imaginary part (a multiple of i, the square root of -1). Complex
numbers can be visualised as arrayed on the complex plane,
with real numbers on the horizontal axis and imaginary numbers
on the vertical axis.
Riemann found that certain complex numbers, when plugged
into the zeta function, produce the result zero. The few zeros
he could calculate lay on a vertical line in the complex plane,
and he guessed that, except for a few well-understood cases,
all the infinity of zeros should lie exactly on this line.
What does this have to do with the primes? If you plot how
many primes exist below a given number (see Diagram above),
what you get is a smooth curve with small wiggles
added--that is, the 1/ln(x) rule, plus deviations.
According to Michael Berry of Bristol University, you can think
of that pattern of deviations as a wave. Just like a sound
wave, it is made up of many frequencies. "And what are the
frequencies?" asks Berry. "They're the Riemann zeros. The
zeros are harmonies in the music of the primes."
Berry isn't speaking in metaphors. "I've tried to play this music
by putting a few thousand primes into my computer," he says
"but it's just a horrible cacophony. You'd actually need billions
or trillions--someone with a more powerful machine should do
Riemann worked out that if the zeros really do lie on the
critical line, then the primes stray from the 1/ln(x) distribution
exactly as much as a bunch of coin tosses stray from the
50:50 distribution law. This is a startling conclusion. The
primes aren't just unpredictable, they really do behave as if
each prime number is picked at random, with the probability
1/ln(x)--almost as if they were chosen with a weighted coin.
So to some extent the primes are tamed, because we can
make statistical predictions about them, just as we can about
But only if Riemann's guess was right. If the zeros don't line
up, then the prime numbers are much more unruly. As Enrico
Bombieri of the Institute for Advanced Study in Princeton
writes on the Clay Institute website
(www.claymath.org/prize_problems/riemann.htm): "The failure
of the Riemann hypothesis would create havoc in the
distribution of prime numbers." And the havoc would spread
further. Hundreds of results in number theory begin, "If the
Riemann hypothesis is true, then . . ."
This is why mathematicians long to prove the hypothesis. But
how do you prove something about an infinity of numbers?
Researchers have used supercomputers to calculate the first
1,500,000,001 zeros above the x-axis, and millions of other
zeros higher up, and so far all of them lie on the critical line. If
just one of them did not, the Riemann hypothesis would be
This is heartening, but no amount of computer hacking can
prove the hypothesis. There are always more zeros to check.
And, cautions Andrew Odlyzko of AT&T Labs, who has
spearheaded the effort to calculate zeros, "number theory has
many examples of conjectures that are plausible, are
supported by seemingly overwhelming numerical evidence, and
yet are false."
Some deeper insight is needed. Early in the 20th century,
mathematicians made a daring conjecture: that the Riemann
zeros could correspond to the energy levels of a quantum
Quantum mechanics deals with the behaviour of tiny particles
such as electrons. Crucially, its equations work with complex
numbers, but the energy of a physical system is always
measured by a real number. So energy levels form an infinite
set of numbers lying along the real axis of the complex
plane--a straight line.
This sounds like Riemann's zeros. The line of zeros is vertical,
rather than horizontal, but it is a simple bit of maths to rotate
it and put it on top of the real line. If the zeros then match up
with the energy levels of a quantum system, the Riemann
hypothesis is proved.
For decades, this idea was only wishful thinking. Then in 1972
came a hint that it could work. Hugh Montgomery, at the
University of Michigan, had found a formula for the spacings
between Riemann zeros. Visiting the Institute for Advanced
Study at Princeton, he ran into physicist Freeman Dyson at
afternoon tea, and mentioned his formula. Dyson recognised it
immediately. It was identical to a formula that gives the
spacings between energy levels in a category of quantum
systems--quantum chaotic systems, to be precise.
Chaos theory applies to physical systems so sensitive to their
starting conditions that they are impossible to predict. In the
Earth's chaotic atmosphere, for example, the tiny draught
caused by the flap of a butterfly's wings can eventually lead
to a tremendous storm. Almost all complicated systems are
The quantum versions of these systems have a jumble of
energy levels, scattered apparently at random but in fact
spaced according to Montgomery's formula. Quantum chaotic
systems include atoms bigger than hydrogen, large atomic
nuclei, all molecules, and electrons trapped in the microscopic
arenas called quantum dots. Could the Riemann zeros fit one
of these quantum chaotic systems?
In the late 1980s, Odlyzko picked an assortment of systems,
and compared their energy levels with the Riemann zeros. In a
discovery that electrified mathematicians and physicists,
Odlyzko found that when he averaged out over many different
chaotic systems, the energy level spacings fitted the Riemann
spacings with stunning precision.
That's still not enough. To prove the Riemann hypothesis,
researchers must pinpoint a specific quantum system whose
energy levels correspond exactly to the zeros, and prove that
they do so all the way to infinity. Which, of all the different
systems, is the right one?
Berry and his colleague Jonathan Keating have made one
suggestion. In a chaotic system, an object usually moves
unpredictably, but sometimes its path will cycle back on itself
in a "periodic orbit". Berry and Keating think that the right
quantum system will have an infinite collection of periodic
orbits, one for each prime number. And last year, Nicholas Katz
and Peter Sarnak predicted that the system should have a
special kind of symmetry called symplectic symmetry.
Both of these clues should help quantum chaologists zero in on
the one system that will prove the Riemann hypothesis. "I
have a feeling that the hypothesis will be cracked in the next
few years," says Berry. "I see the strands coming together.
Someone will soon get the million dollars."
The winner could well be Alain Connes, a mathematician based
at the Institute of Advanced Scientific Study in
Bures-sur-Yvette, France. Connes has a startlingly direct
approach to the problem: create a system that already
includes the prime numbers. To understand how, you have to
imagine a quantum system not as a particle bouncing around
an atom, say, but as a geometrical space. It sounds odd, but
it represents one of the weird things about quantum systems:
they can be two or more things at once.
Like Schrödinger's cat, which is a peculiar mixture of dead and
alive, any quantum object can find itself in a "superposition" of
different states. To characterise this messy existence,
physicists use what they call a state space. For each kind of
possibility (say "alive" and "dead"), you draw a new axis and
add a dimension to the space. If there are just two possible
states, as is the case for Schrödinger's cat, the space is two
dimensional, with three states it is three dimensional, and so
Then in the Schrödinger's cat space, you would mark a cross
one unit along the x-axis to represent a fully alive cat.
Similarly, a stone dead cat would be one unit up the y-axis,
and a part-alive, part-dead cat would appear somewhere
along an arc between these points.
The "shape" of the space affects how the state moves around
in it, and therefore how the system works, including the way
its energy levels are arrayed. This depends not just on the
number of dimensions, but also on the geometry of how they
are stuck together.
Connes decided to build a quantum state space out of the
prime numbers. Of course, the primes are a bunch of isolated
numbers, nothing like the smooth expanses of space in which
we can measure things like angles and lengths. But
mathematicians have invented some bizarrely twisted
geometries that are based on the primes. In "5-adic"
geometry, for example, numbers far apart (in the ordinary
way) are pulled close together if they differ by 5, or 15, or
250--any multiple of 5. In the same way, 2-adic geometry
pulls together all the even numbers.
To put all the primes in the mix, Connes constructed an
infinite-dimensional space called the Adeles. In the first
dimension, measurements are made with 2-adic geometry, in
the second dimension with 3-adic geometry, in the third
dimension with 5-adic geometry, and so on, to include all the
Last year Connes proved that his prime-based quantum
system has energy levels corresponding to all the Riemann
zeros that lie on the critical line. He will win the fame and the
million-dollar prize if he can make one last step: prove that
there aren't any extra zeros hanging around, unaccounted for
by his energy levels.
That last step is a formidable one. Has Connes simply replaced
the Riemann hypothesis with an equally difficult question?
Some experts advise caution. "I still think that some major new
idea is needed here," says Bombieri.
Berry, for his part, doesn't flinch at the mathematical
peculiarity of Connes's system. "I'm absolutely sure that if he's
right, someone will find a clever way to make it in the lab.
Then you'll get the Riemann zeros out just by observing its
Berry and Keating are now turning around this connection with
physics, using mathematics based on the Riemann zeta
function to predict the behaviour of chaotic systems. Most
models of quantum chaos are complicated and difficult to
calculate. The Riemann zeros, by comparison, are easy to
compute. "We always test our formulae on the Riemann zeta
function to see if they work," says Keating.
If Connes or one of the physicists proves the Riemann
hypothesis using a quantum system, the link will be firmly
established. Then, Berry predicts, the field will blossom. Using
the mathematics of the zeta function, scientists will be able to
predict the scattering of very high energy levels in atoms,
molecules and nuclei, and the fluctuations in the resistance of
quantum dots in a magnetic field.
And it turns out that the same mathematics applies to any
situation where waves bounce around chaotically, including
light waves and sound. So the performance of microwave
cavities and fibre optics could be improved, and the acoustics
of real concert halls might profit from the music of the primes.
Even so, it is mathematics that will gain the most. "Right now,
when we tackle problems without knowing the truth of the
Riemann hypothesis, it's as if we have a screwdriver," says
Sarnak. "But when we have it, it'll be more like a bulldozer."
For example, it should lead to an efficient way of deciding
whether a given large number is prime. No existing algorithms
designed to do this are guaranteed to terminate in a finite
number of steps.
Proving the Riemann hypothesis won't be the end of the story.
It will prompt a sequence of even harder, more penetrating
questions. Why do the primes achieve such a delicate balance
between randomness and order? And if their patterns do
encode the behaviour of quantum chaotic systems, what other
jewels will we uncover when we dig deeper?
Those who believe that mathematics holds the key to the
Universe might do well to ponder a question that goes back to
the ancients: What secrets are locked within the primes?