Prime time

                  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?