Mathematicians dream of proving Riemann's Hypothesis.by Karl Sabbagh / January 20, 2002 / Leave a comment
Over the last century or so, maths became “modern.” It has reached new heights of abstraction and provided tools for the revolution in physics and cosmology that came to the public’s attention with Einstein. Modern mathematics now routinely handles concepts such as infinities of different magnitudes and has invented new numbers which lie nowhere on the line from minus infinity to plus infinity. It has devised closed shapes with finite areas but infinite perimeters. Techniques which at first seem to have no useful function turn out decades later to provide tools for use in the real world. But the essential joy of maths has remained unchanged. When a mathematician proves a hypothesis, he or she has discovered something which is true forever.
Living on a planet that is running out of unexplored territory, the mathematical explorer can never run out of discoveries to savour. And the mathematical world is now at a significant moment: one of its most important unsolved problems may be on the brink of solution.The problem has been around since 1859 and every pure mathematician has longed for it to be resolved. In fact, it may be insoluble, but even knowing that for certain would be a major advance.
This is not a problem that is important in the way that splitting the atom was, or unravelling the genetic code. But last year a US foundation thought the problem merited a prize of $1m to the first person to provide a proof. In fact, the Clay Foundation offered seven prizes of $1m each for proofs of what it considered to be the most significant maths problems. (Some of them sound deceptively easy-“prove that P = NP” for example.) But the most important of these problems, and the most difficult judging by the effort being put into it, is the Riemann Hypothesis.
In 1859, a German mathematician called Bernhard Riemann, a “timid diffident soul with a horror of attracting attention to himself,” published a paper that drew more attention to him than to almost any other mathematician in the 19th century. In it he made an important statement: the non-trivial zeros of the Riemann zeta function all have real part equal to H.
That is the Riemann Hypothesis: 15 words encapsulating a mystery at the heart of our number system. If you read the statement with no mathematical knowledge, you could still ask a meaningful question about it. You could work out that there are things called “non-trivial zeros of the Riemann zeta function” and that something is said to be true about all of them. They all have “real part equal to a half.” The statement is therefore testable. If we call “non-trivial zeros of the Riemann zeta function” snarks and anything with “real part equal to a half” a boojum, those people who know what the Riemann zeta function is need only take the snarks and do something with them that will prove that they all have boojums. They will also win the prize if they disprove the hypothesis, by finding one snark that does not have a boojum.
For many mathematicians working on it, $1m is less important than the satisfaction that would come from finding a proof. Throughout my researches among the mathematicians’ tribe (I have interviewed 30 in the past year), Riemann’s Hypothesis was often described to me in awed terms. Hugh Montgomery of the University of Michigan said this was the proof for which a mathematician might sell his soul. Henryk Iwaniec, a Polish-American mathematician, sounded as if he were already discussing terms with Lucifer. “I would trade everything I know in mathematics for the proof of the Riemann Hypothesis,” he said. “It’s gorgeous stuff. I’m only worried that I’ll be unable to understand it. That would be the worst…”
What lies behind this 15-word statement is a question about prime numbers, the building blocks of our number system. Despite the self-contained abstraction of pure maths, prime numbers appear to possess a concrete “existence.” “Primes are like things you can touch,” Yoichi Motohashi, a Japanese mathematician at Nihon University, told me. “In mathematics most things are abstract but I have some feeling that I can touch the primes, as if they are physical particles.”
Below is an incomplete list of integers (whole numbers). Among them, in bold, are the primes:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45…
A prime number cannot be divided by any other whole number (except 1) without leaving a remainder. In their indivisibility, they are like atoms and the other numbers, known as composites, are like molecules. You cannot “split” 19 into the product of smaller numbers, like you cannot split an atom into smaller atoms. Another fact about the primes was proved by Euclid-they go on forever. However far you count, you will always come across primes.
Ever since people started counting, the nature of the primes has aroused interest. Out of one simple property of these numbers has emerged an edifice of theory, leading to fields of maths which require years of training to grasp.
There’s also a lot we do not know about primes. We do not know what determines their distribution among the integers. Sometimes, as you count up from one, two of them come along together, separated by one integer; at other times you could go through hundreds or thousands of numbers without finding a single prime. As mathematicians started to analyse the statistics of the distribution of the primes in more detail, they uncovered a hidden structure that they could describe but not explain. The Riemann Hypothesis would provide an explanation for that structure.
But a proof of the Riemann Hypothesis would be far more significant than providing one piece in a jigsaw puzzle. It would form the final piece for hundreds of puzzles, all incomplete. Number theory, the field that investigates integers, is full of conjectures that start, “if the Riemann Hypothesis is true, then…” So strong is the feeling among mathematicians that the Riemann Hypothesis is true, but too hard to solve, that they have rushed ahead with other theorems that depend on the hypothesis.
The central element in the Riemann Hypothesis is a mathematical expression called the Riemann zeta function. A “function” in mathematics is like a black box into which you put a number at one end and a different number comes out the other. What is important for the Riemann zeta function is that, with certain inputs, you get zero out at the other end. This can be illustrated with a simple function, such as x2-4. If you express this as an equation and put 2 “in one end”-substitute it for x-you get zero out of the other (22-4=0). But where the function x2-4 has only two “zeros” (x=2 and x=-2) the hugely more complex Riemann zeta function has an infinite number. All these zeros, if plotted on a special kind of graph, are believed to lie in a straight line.
Riemann calculated the positions of the first few zeros-with difficulty because it was a laborious task-and they did lie on the expected line. But calculate as many points as you like, you will never be able to demonstrate that all the points lie in a straight line, since there is an infinity of them. This has not stopped mathematicians calculating zeros into the billions using computers, and no one has yet found a zero that is off the line. But demonstration is not proof. One famous conjecture, called Mertens’ Conjecture, was true for all numbers up to 1,000,000,000,000,000,000,00 0,000,000,000 but false after that. Mathematicians can only move ahead cautiously, knowing that their efforts are built on sand if they rely only on confirming examples. That is why a proof of the Riemann Hypothesis is so desirable.
Anyone who thinks that mathematicians are good at mental arithmetic should meet the French-American mathematician, Louis de Branges. We were discussing the idea that mathematicians did their best work while young, and I asked him when he had made a particular insight. “Let’s see,” he said, “I was born in 1932 and I made the discovery in 1984. So was I over 50? How old was I…?” He wrestled with the problem and then gave up.
I met de Branges because he said he was putting the final touches to a proof of the Riemann Hypothesis. It took only two or three phone calls to other mathematicians to realise that no one took that seriously. This wasn’t the first time de Branges had claimed to be near a proof of Riemann. In their view, there was “no chance” that de Branges had a proof; he was “always telling people he had a proof and it was always riddled with errors”; he was “working in the wrong area,” and so on.
De Branges has a personality that many people find brusque, obsessive and stubborn. You can see how colleagues might be deterred from giving his work on the Riemann Hypothesis the attention he believes it deserves. But their scepticism made me want to find out more about de Branges, as a way of understanding what the Riemann Hypothesis was and what mathematicians do when they are trying to prove it. Here was a man who had been thinking about the hypothesis for 20 years. Surely he would know it as well as any mathematician, even if he was barking up the wrong tree. The history of mathematics is littered with false proofs. Every major unsolved problem leads to swathes of amateur attempts landing on the desks of professional mathematicians. But de Branges had one factor which suggested that he was not a crank. He had proved the Bieberbach Conjecture in 1984, 68 years after it was formulated.
The Bieberbach Conjecture is even more difficult to explain than the Riemann Hypothesis (it deals with “normalised injective holomorphic functions”). But it is enough to know that de Branges’ achievement earned him the right to be taken seriously as a mathematician. “When I understood it,” says Xian Jin Li, who studied under de Branges, “I appreciated how beautiful his original proof really is.”
Beauty is a concept used by mathematicians (and other scientists) when they find something that combines truth and simplicity. Like the use of the word in everyday life, you know it when you see it. It is possible to distinguish superficial from less superficial beauties in maths, just as you can between trite and subtle harmonies in music. But in Wittgensteinian terms: one cannot explain, one can only show.
Yet something went wrong when de Branges produced his “beautiful solution.” It is difficult to follow the events that followed his announcement of his proof of the Bieberbach Conjecture. In de Branges’ moment of triumph, he believed that credit was snatched away by jealous colleagues. Nursing his wounds, he turned to the Riemann Hypothesis and over the last 15 years has let it be known that he is near to a proof. He has too much self-belief to consider that he might be wrong and has told me that 2002 will be the year he really offers a watertight proof of the Riemann Hypothesis.
The last great mathematical problem to be solved was Fermat’s Last Theorem. This problem was much easier to state than the Riemann Hypothesis, but not much easier to prove. The theorem states that there is only one value of n for which whole numbers x, y, and z, can be found to satisfy the expression xn+yn=zn, and that value is 2. Like the Riemann Hypothesis, the problem conceals more than it reveals. As the successful proof showed, Fermat’s Last Theorem identified a route into fields of mathematics that were not obviously connected with the sums of powers of whole numbers. Attempts to prove it led to most of modern algebra. It is often the difficulty of proving an apparently simple theorem that demonstrates its importance to the wider field.
After decades of solitary calculation, the English mathematician Andrew Wiles published a correct proof, which runs to more than 100 pages. But there were some mathematicians who resented the fact that Wiles carried out his work in secrecy. More than one of them wasted years going down blind alleys which Wiles had already explored and dismissed. Now, with the stakes even higher, current work on the Riemann Hypothesis raises the same fears. Whatever mathematicians say about the pure joys of doing mathematics, there is no denying the human pleasure that comes from achieving a goal before your colleagues.
Atle Selberg, a distinguished Norwegian number theorist in his 80s, is believed by many to be pursuing the Riemann Hypothesis, after working on it for the last 60 years and making important contributions to the theory of primes. He has demonstrated that a positive proportion of the zeros of the Riemann zeta function do lie on the required straight line. But he has never in public said that he was near a proof. When I asked him whether he was looking, he replied: “my mind is still reasonably alert.”
Henryk Iwaniec is also thought capable of proving the Riemann Hypothesis. He is an example of a “younger mind” that has made several minor breakthroughs already. (Mathematics’ “Nobel Prize,” the Fields medal, goes only to mathematicians under 40.) Iwaniec sees nothing wrong in keeping promising results to himself: “collaboration is a nice thing, but when it comes to the best work, this is too good to share.” Such are the delights of the field.
Andr? Weil, sister of Simone Weil, compared the mathematical enterprise to sex: “Every mathematician worthy of the name has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously. Unlike sexual pleasure, this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do so at will, unless perhaps by dogged work.” For many mathematicians the pleasure of practising mathematics outweighs almost any other mental or physical pleasure. Andrew Granville, an English mathematician working at the university of Georgia, told me: “It’s the beauty that justifies things… you just have to do it.”
Most mathematicians believe they are making real discoveries about real objects in a real world. Not our real world, nor a world whose existence can be conclusively demonstrated. But a world of ideas that appear more real than the ideas of ethics, politics or aesthetics. There is no ambiguity in a mathematical discovery. You cannot argue with a mathematical proof in the way that people argue about other ideas.
It is possible to go further and, as Pythagoras did, regard mathematics as no less than the secret of the universe. Scientists from Galileo’s time onward discovered not only that physical phenomena appear to be perfectly captured by mathematics, but that the world seems to be mathematical in character. Galileo called maths, “the language of nature” and Sir Arthur Eddington said that a study of physics gives us a “knowledge of mathematical structures.” The physical universe, for all its diversity, can be described with precision in terms of a relatively small group of equations. And where the physicists led the biologists followed with their equations for evolution. The mathematical picture of nature makes possible such things as electric light, space exploration and the destruction of Hiroshima. (Many people working on the Manhattan project used sophisticated mathematics to help work out the behaviour of sub-atomic particles.)
And yet, mathematicians are not, for the most part, people who do maths to learn about the outside world. They do maths to learn about maths. When Riemann first formed his hypothesis he gave no thought to whether it would improve a future generations of mobile phones. The surprising thing is, it might, through a new method of encrypting communications. But few mathematicians have that as their aim, although some are paid by companies that will benefit from such encryption. What drives them is the pleasure of discovering something new about the number system, in the same way that a scientist discovers new facts about the physical universe.
The ideas of theoretical physics start with real objects, from atoms to galaxies; the ideas of mathematics start with concepts and can lead to descriptions of reality. But are mathematical objects “out there” like astronomical objects are? Some philosophers argue that mathematics is the creation of mind. We define certain concepts-number, or set-and then define certain operations that can be performed with them according to certain rules. All the truths that follow are the logical consequences of these definitions. A change of definitions produces a change in the resulting truths: witness Alice’s version of multiplication as she falls down the rabbit-hole. If mathematics is the product of thought, and reality mathematical in character, is mind the ultimate reality?
To some mathematicians that philosophical distinction is unimportant. How the entities referred to in mathematics-numbers, sets, operations, functions-constitute the solid objects of the world ceases to matter once you have experienced them first hand. Matti Jutila from Finland said, “I sometimes have the feeling that the number system is comparable with the universe that the astronomer is studying…The number system is something like a cosmos.”
If you work with integers, discovering properties that can be demonstrated to other mathematicians, exploring the relationships between prime numbers, or squares and higher powers, they acquire a reality which is hard to deny. You make observations, form hypotheses, and can test them by gathering data, as observations of the galaxies allows cosmologists to test theories about the age of the universe.
Leopold Kronecker, a 19th-century German mathematician, said, “God gave us the positive integers; man did all the rest.” This is a compromise that some mathematicians would accept. There is something so indisputable about the numbers that a suggested way of communicating with alien intelligences is by sending them sequences of prime numbers in the belief that they would recognise them. The rest, the theorems, hypotheses and conjectures of number theory, including the Riemann Hypothesis, may be the work of man.
For other mathematicians, theory is pointless. “I never think about such a thing, never,” Peter Sarnak said. “We know what we’re after, we’re working people. The philosophy comes when you’re looking back, taking stock. When you’re out there in the front line, you can’t start worrying about whether numbers are real.” When I asked Alexander Ivic, a gravel-voiced Croatian with several discoveries under his belt, whether mathematics is discovered or invented, he said: “I don’t deal with philosophical questions. They are too unnerving, so to speak.”
I believe that there is something “out there”; that mathematics is a description of something external. It is hard to think of an endeavour that has the consistency and universal acceptability of mathematics-what has been called “the unreasonable effectiveness” of mathematics is difficult to deny.
Since first meeting de Branges, I have been in touch with him several times and on each occasion his proof has been imminent. In fact, he once thought up a crucial ingredient while waiting for my train to arrive at the station near his French summer home. But even if he does publish a correct proof, there may be no mathematician willing to read it. An important mathematical proof cannot be understood by glancing at a few pages of symbols. It can run to hundreds of pages of dense, specialised mathematics. You must believe that it has a chance of being correct to commit the time to read it. De Branges may end up in the strange position of believing that he is the only person who knows the proof of the Riemann Hypothesis.
But for mathematicians like Charles Ryavec, that solitary satisfaction would not be enough. “I was talking to a guy once and I said, ‘suppose you’re the last person on earth: are you going to work on the Riemann Hypothesis?’ We kind of agreed no. If you’re the only one left, what are you supposed to do? Turn off the lights? Shut the door? Do the Riemann Hypothesis?-it’s not going to be that important.”
“But what if you were among the last five people on earth, all of them mathematicians?” I asked.
“With five or ten, I probably would,” he said.