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

Cambridge, Massachusetts.

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

it."

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

coin tosses.

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

killed.

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

mechanical system.

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

chaotic.

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

on.

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

prime numbers.

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

spectrum."

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?