## Riemann Hypothesis Has Finally Been Solved Assignment

Two years ago I wrote a book about the Riemann Hypothesis (for an account of the hypothesis see A.W. Moore’s article in this issue). The proof of it is something that every mathematician would love to discover or – very much second best – see someone discover. One of the people I interviewed was Louis de Branges, a Franco-American mathematician at Purdue University in Lafayette, Indiana, with one significant proof already under his belt. De Branges was convinced that a mathematical field in which he was the acknowledged specialist would lead to a proof of the hypothesis. I have stayed in touch with him, and earlier this year, he told me he was putting the finishing touches to a proof he has been working on for 25 years. On 28 April this proof was published on the internet for other mathematicians to see and criticise: www.math.purdue.edu/~branges.

There is no evidence that, so far, any mathematician has read it: de Branges and his proof appear to have been ostracised by the profession. I have talked to a number of mathematicians about him and his work over the last few years and it seems that the profession has come to the view that nothing he does in this area will ever bear fruit and therefore his work can be safely ignored. It may be that a possible solution of one of the most important problems in mathematics is never investigated because no one likes the solution’s author.

De Branges’s paper was slipped onto the internet without a fuss. Had he been any other mathematician, there would have been rumours beforehand. Over the last three years I have got to know a number of the key men – they are all men – who work on the Riemann Hypothesis, and although each of them keeps his cards close to his chest, they are all desperate to get a look at the other fellow’s hand. I spoke to about twenty of the mathematicians most likely to prove the hypothesis, and if any of them was within reach of a proof, the others would be agog to see what he was doing. Except, of course, in the case of de Branges.

De Branges wrote to me in February this year telling me that he was ready to publish his proof. He doesn’t use email, and writes all his letters by hand.

The final form of the proof of the Riemann Hypothesis will follow closely the treatment of Hilbert spaces of entire functions which I discovered in the years 1957-62 and which was published as a book in 1968. An elegant proof is given which should cause no difficulty for verification. A reader does, however, need to acquire a broad knowledge of these spaces to read the argument.

He mentioned several mathematicians he thought would have this broad knowledge, and went on:

I will give a copy of the manuscript to Paul Malliavin as editor of the

Journal of Functional Analysis. But there is no certainty that he will consider the paper for publication over the next few months. Each of them will certainly say that it contains material relevant to his special interests. They will certainly guard themselves against any assertion that the argument is correct.

When de Branges told me his proof was complete I suspected that his paper would be dismissed without being read. Sure enough, in early May, after the internet publication, when reporters from *New Scientist* and *Nature* started to look into it and to consider whether this really was the most important mathematical discovery of the last hundred years, their own mathematical contacts assured them that it could safely be ignored. But none of these mathematicians claimed to have actually read de Branges’s paper.

The first thing to say about this odd situation is that de Branges is not a crank. Most mathematicians working on this problem receive a regular stream of alleged proofs from people with little or no grasp of number theory or complex analysis, the most likely fields from which a proof was expected to emerge. Most of these ‘proofs’ go into the bin unread. But on the basis of track record, ability and originality of thought, de Branges is in a very different category.

He may not be a crank, but he is cranky. ‘My relationships with my colleagues are disastrous,’ he told me. And he does seem to have left a trail of disgruntled, irritated and even contemptuous colleagues behind him if only because he makes no concessions to students and colleagues who are not familiar with the field in which he works. It may be a field largely of his own devising, but it makes a genuine contribution to pure mathematics. When he’s fortunate enough to have students to teach he makes them work their way through a series of extremely tough exercises and sees no reason to make it easy for them.

He is a person of strict routine: it’s the only way he can create the right conditions for the mathematical thought processes which take up most of his waking life. Adherence to rules is very important. When I was walking with him in France, he remonstrated with me because I stepped on a zebra crossing when two cars were at least a hundred yards away. ‘The cars have to stop if you are on the crossing,’ he said, ‘and one of them might have driven into the back of the other.’ He only ever watches one TV programme, the CBS news. ‘We cannot afford the time for more television,’ he says.

He is also disarmingly honest. He’s even honest about how honest he is:

I differ from other mathematicians in that I seem to have a deep honesty that other people sometimes don’t have, and it’s rather curious because I certainly didn’t have that intention as a young man. I think I was by nature somebody that would easily cut a corner – especially if I didn’t think it was very important – not for any real advantage, but I would choose to do so. But the way my life has evolved, against my own inclinations, I have turned out to have an unusual probity.

People who keep telling you how modest they are are not usually modest at all. De Branges isn’t like that: he *is* honest. There have been occasions when he has told me things that other people would think twice about revealing. ‘My mind is not very flexible,’ he once said:

I concentrate on one thing and I am incapable of keeping an overall picture. So when I focus on the one thing, I actually forget about the rest of it, and so then I see that at some later time the memory does put it together and there’s been an omission. So when that happens then I have to be very careful with myself that I don’t fall into some sort of a depression or something like that. You expect that something’s going to happen and a major change has taken place, and what you have to realise at that point is that you are vulnerable and that you have to give yourself time to wait until the truth comes out.

This kind of single-mindedness can be seen in people with Asperger’s Syndrome.

Occasionally he has surprised me by talking about his personal life. For example, on one occasion he embarked on the story of his first marriage and ended up telling me how he likes to whistle tunes in the street. It provides a good example of the rather formal way he speaks:

I’d married a student from Bryn Mawr College, and all of a sudden she just left, asking for a very substantial amount of money which I didn’t in any way contest. And then staying around in Lafayette for about ten years, that greatly created a circle of opposition within the community, because, you see, I was a person that was seen as being in the wrong by my colleagues, and also by the community.

The divorce was seen as a criticism of myself, of my performance. To give you an example of that, my wife sang in an organisation called the Bach Chorale, and I was seen by musical people as being somebody that would be against the musical traditions or the arts. Well, this is a curious thing: I happen to be very musical, I simply don’t have a musical education because of the war years. My musical qualities are expressed by my whistling. Usually, you know, when you whistle you disturb people, and I apologise for that, but people like the things that I whistle. They say: ‘Oh, that’s a nice melody, I like that.’ It happened when I was going to fetch you at the station, some young lady said: ‘Yes, I like to hear that.’ I’m sure that my musical quality is much greater than that of the girl who divorced me. I used to sing also in a choir, so I can have a good voice. My speaking voice is rather flat, but my singing voice is good.

Whatever personal eccentricities de Branges might have, it’s hard to believe they would be enough to make mathematicians who are desperate for a proof of the Riemann Hypothesis reject the possibility that he might now have one.

Yet it has been dismissed as ‘probably cobblers’. One reason is that mathematicians seem to think that de Branges has claimed on several previous occasions to have proved the Riemann Hypothesis and been in error. ‘He has made something of a tradition, I’m told, of emailing colleagues every September with a new proof he worked up over the summer,’ another mathematician told me. Successive versions of de Branges’s paper were posted on the internet as his ideas evolved. But it is unlikely that he has ever emailed any colleagues anything. He is in contact with very few of them and, in any case, doesn’t use email.

De Branges has certainly made errors in the past, but it is difficult to find a mathematician who hasn’t. ‘The first case in which I made an error was in proving the existence of invariant subspaces for continuous transformations in Hilbert spaces,’ he told me. ‘This was something that happened in 1964, and I declared something to be true which I was not able to substantiate. And the fact that I did that destroyed my career. My colleagues have never forgiven it.’

Since then, de Branges has on one occasion believed that he had a finished proof of the Riemann Hypothesis, until an error was pointed out, and he has also believed himself to be near a proof on several occasions before himself discovering a mistake. But mathematicians are surely expected to show a degree of objectivity in assessing their colleagues’ work. Even if de Branges were the error-prone sociopathic curmudgeon some believe him to be, is that really enough to stop anyone even considering the possibility of a proof of the Riemann Hypothesis?

Maybe de Branges just isn’t a very good mathematician. But it is generally agreed that he did solve another important mathematical problem, the Bieberbach Conjecture, in 1985. Not only that: there are uncanny similarities between the initial reaction of other mathematicians to his claim to have proved the Bieberbach Conjecture then, and the unwillingness now to consider that he might have proved the Riemann Hypothesis. ‘It would be easy to dismiss de Branges as a crank,’ one mathematician wrote on the internet, ‘but he has earned the right to a hearing because the early dismissals of his work on the Bieberbach Conjecture turned out to be wrong.’

‘I am sure that Louis de Branges’s many "wrong” proofs of the Riemann Hypothesis and other conjectures are as chock-full of brilliant ideas as is his proof of Bieberbach,’ another wrote.

A third, in a festschrift to celebrate de Branges’s Bieberbach Conjecture proof, said: ‘In March of 1984 the message began to travel. Louis de Branges was claiming a proof of the Bieberbach Conjecture. And his method had come from totally unexpected sources: operator theory and special functions. The story seemed fantastic at the time, but it turned out to be true.’

‘Bieberbach was a tremendous achievement, there’s no question about it,’ Peter Sarnak of the Institute for Advanced Studies says. ‘Louis de Branges hit the big time there, really. It was a great problem . . . and his solution was absolutely brilliant, really brilliant.’ But Sarnak is one of many who dismiss his Riemann Hypothesis proof.

Atle Selberg, one of the greatest pure mathematicians of modern times, said to me:

The thing is it’s very dangerous to have a fixed idea. A person with a fixed idea will always find some way of convincing himself in the end that he is right. Louis de Branges has committed a lot of mistakes in his life. Mathematically he is not the most reliable source in that sense. As I once said to someone – it’s a somewhat malicious jest but occasionally I engage in that – after finally they had verified that he had made this result on the Bieberbach Conjecture, I said that Louis de Branges has made all kinds of mistakes, and this time he has made the mistake of being right.

De Branges is now claiming to have solved another, far more significant problem than the Bieberbach Conjecture, again from ‘totally unexpected sources’, and again most people are treating the story as fantastic. Will the mathematical community again come to accept the proof?

It seems unlikely, since there is no one who has read the 121-page paper all the way through who is competent to judge it. Because de Branges’s proof uses mathematical tools in which he is one of the few experts, the amount of study required even to become familiar with those tools before embarking on reading the paper seems too great for anyone to commit the time. Even the few people who know and understand de Branges and his method see it as a daunting task. Nikolai Nikolski helped with the validation of the Bieberbach Conjecture proof, a task that took a team of mathematicians at the Steklov Institute in Leningrad several months. ‘The Riemann Hypothesis is much more complicated than the Bieberbach Conjecture,’ Nikolski told me.

So you have to be more enthusiastic if you want to validate the proof. You need to have a team of really enthusiastic high-level people. De Branges found in the middle of the 1980s the only place in the world where there were some curious people who just love to solve complicated problems and who were ready to spend a half a year on it. He has asked me several times if it’s possible to organise some people to do the same thing with the Riemann Hypothesis. I love him, so I said to him: ‘Yes, if you have a very huge grant, probably not so huge as in America, to pay, for instance, the same place in Petersburg.’

But there are plenty of influential mathematicians who just think the whole process would be a waste of time. Brian Conrey, the director of the American Institute for Mathematics, who is developing his own ideas for a proof of the Riemann Hypothesis, is insistent: ‘I just *know* it can’t come out of de Branges’s approach,’ he said. ‘It’s the wrong theory.’ But he added a complimentary afterthought: ‘If only he was to market his results for what they are – it *is* a very beautiful theory.’

Bela Bollobas, a fellow of Trinity College, Cambridge who teaches at the University of Memphis is less dogmatic:

De Branges is undoubtedly an ingenious mathematician, who established his excellent credentials by settling Bieberbach’s Conjecture . . . Unfortunately, his reputation is somewhat tainted by several claims he made in the past, whose proofs eventually collapsed. I very much hope that this is not the case on this occasion: it is certainly not impossible that this time he has really hit the jackpot by tenaciously pursuing the Hilbert space approach. Mathematics is always considered to be a young man’s game, so it would be most interesting if a 70-year-old mathematician were to prove the Riemann Hypothesis, which has been considered to be the Holy Grail of mathematics for about a hundred years.

When I visited de Branges in his flat near Paris in May, he did not behave like a man who was in sight of a million-dollar prize. But this was not because of any doubts about his proof. ‘The proof is there,’ he said, ‘but it’s just part of a longer paper on the zeta functions. That’s the important work. It’s a theory that could lead to a new understanding of quantum physics, for example, since the way I approach the subject uses a type of mathematics – spectral theory – that seems to underlie the behaviour of atoms.’

I asked him how he felt, now that he ‘knew’ the Riemann Hypothesis was correct, expecting some expression of satisfaction, or even exhilaration. ‘It’s a question of sanity,’ he said. ‘When you have a wife who doesn’t understand what you do and just wants you out of the house’ (his former wife – he is now happily married); ‘when you have a mother who comes to live with you to look after you and can’t understand what you are doing; when you have colleagues who ignore or dismiss your work . . .’ His voice tapered off. ‘I just hope someone doesn’t come along now with an elementary proof of the Riemann Hypothesis.’

I was puzzled by this. It can’t have been a matter of priority, since his proof is now out on the internet, dated 28 April, and if it is verified he will get the credit for it. But it turned out he was worried that were someone else to prove the hypothesis without using his broader theory of zeta functions, his life’s work would be sidelined as people focused on the other proof and ignored the new insights he felt he had achieved. ‘That would be a disaster,’ he said.

Perhaps one day a young mathematician steeped in de Branges’s theory of Hilbert spaces of entire functions will pick up his paper and begin to work through it. Or perhaps, as the news spreads that the entire mathematical profession is turning its back on what could be the most important development in the last hundred years of mathematics, one or two practitioners will be shamed into reading through de Branges’s proof, just in case he really has cracked an important problem for the second time in his working life.

One of the most important problems in mathematics - the Riemann Hypothesis - is said to have finally been solved by a Nigerian professor.

Dr Opeyemi Enoch claims he made a key breakthrough in 2010 which later enabled him to solve the puzzle, which is one of the seven Millennium Problems in Mathematics.

These seven puzzles were set by The Clay Mathematics Institute in 2000 and, once confirmed, the organisation is expected to reward Dr Enoch with a $1 million (£658,000) prize for his discovery.

One of the most important problems in mathematics – the Riemann Hypothesis – has been solved by Nigerian professor, Dr Opeyemi Enoch (pictured above). Dr Enoch claims he made a key breakthrough in 2010 which later enabled him to solve the puzzle, which is one of the seven Millennium Problems in Mathematics

The Riemann Hypothesis was proposed by mathematician Bernard Riemann in 1859 and concerns the distribution of prime numbers.

It has become arguably the most famous problem in mathematics, since Fermat's Last Theorem was solved in the 1990s.

At its most simple, the distribution of prime numbers among all others doesn't follow a regular pattern.

However, Riemann noticed that the frequency of prime numbers is very closely related to the behaviour of an elaborate function called the Riemann Zeta function.

The hypothesis asserts that all solutions of the equation ζ(s) = 0 lies on a certain vertical straight line, according to the Clay Mathematics Institute.

While this has been checked for the first 10,000,000,000 solutions, it is only now that a 'proof' explaining their distribution beyond this has been found.

The Riemann Hypothesis was proposed by mathematician Bernard Riemann in 1859 (his working is pictured) and concerns the distribution of prime numbers. It has become arguably the most famous problem in mathematics, since Fermat's Last Theorem was solved in the 1990s

However, The Clay Mathematical institute has neither confirmed nor denied that Dr Enoch has officially solved the problem, simply saying it does not comment on solutions to the Millennium Problems.

This has led to critics claiming the story is a hoax and MailOnline has contacted the professor for more information.

Dr Enoch, who teaches at the Federal University of Oye Ekiti (FUOYE) in Nigeria, said he was motivated to solve the 156-year-old problem because of his students.

He told the BBC that they wanted him to make money from the internet.

'The motivation was because my students trusted that the solution could come from me - not because the financial reward and that was why I started trying to solve the problem in the first place,' he said.

### THE MILLENNIUM PRIZE PROBLEMS

The Millennium Prize Problems were launched on 24 May, 2000.

They include seven problems considered by the Clay Mathematics Institute to be 'important classic questions that have resisted solution over the years'.

These include: P versus NP, The Hodge conjecture, The Poincaré conjecture, The Riemann hypothesis, Yang–Mills existence and mass gap, Navier–Stokes existence and smoothness and The Birch and Swinnerton-Dyer conjecture.

The full details of each are available from the institute's website.

The first person to solve each of the problems will receive $1 million (£658,000).

The professor presented his proof on 11 November during the International Conference on Mathematics and Computer Science in Vienna, Nigerian news site Vanguard reported.

A statement from the university said that having started investigating the problem, Dr Enoch 'went on to consider and to correct the misconceptions that were communicated by mathematicians in the past generations, thus paving way for his solutions and proofs to be established.

'He also showed how other problems of this kind can be formulated and obtained the matrix that Hilbert and Poly predicted will give these undiscovered solutions.

'He revealed how these solutions are applicable in cryptography, quantum information science and in quantum computers.'

Despite his achievement, the academic said some people have been critical by asking "If this man can solve the Riemann problem...why should he not be able to provide solutions to Nigeria's problems?" Dr Enoch said.

'Some guys celebrated it, some criticised it- saying what has that got to do with putting food on the tables of Nigerians.'

Dr Enoch has previously designed a prototype silo for poor farmers and is working on how to protect oil pipelines from vandalism as well as mathematical approaches to climate change.

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