Fame and fortune await the person who cracks the
greatest problem in mathematics.
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?
Erica Klarreich is a mathematician and science writer based in Santa Cruz, California
The zeta function (above) holds inside it the secrets of the primes. How come? Like any function, all it does is turn one number into another number. If n is 3, for example, you add up the infinity of terms in the formula to find out that (3) is roughly 1.2. As the formula shows, the zeta function can also be written as a product of infinitely many terms, each based on one prime number.
The true significance of this function emerges if you feed it complex numbers such as 24 + 13i — combinations of ordinary, real numbers and so-called imaginary numbers (where i is the square root of -1). Although they may sound abstruse, complex numbers are used to simplify practical calculations in everything from engineering to quantum mechanics. By plotting out real parts along the x-axis and imaginary parts along the y-axis, complex numbers can be visualised as points on a two-dimensional plane (right).
For certain complex numbers, the zeta function is zero. All the known "zeros" lie along a line in the complex plane, with real parts equalling 1/2. Riemann's hypothesis is that every zero lies on this line. If they do, Riemann proved, the prime numbers must show up as if they are picked at random, but still following an underlying distribution.
Many other investigations in number theory also depend on the truth of the hypothesis, which is why mathematicians long to pin down the zeros of zeta.
From New Scientist - November 11, 2000, p. 32.