After Sir Michael Atiyah’s presentation of a claimed proof of the Riemann Hypothesis earlier this week at the Heidelberg Laureate Forum, we’ve shared some of the immediate discussion in the aftermath, and now here’s a round-up of what we’ve learned.

Firstly, it’s become clear that the work presented by Atiyah doesn’t constitute a proof of the Riemann Hypothesis, so the Clay Institute can rest easy with their 1 million dollars, and encryption on the internet remains safe. The argument Atiyah put forward rests on his function $T(s)$ having certain properties, and many have concluded that no function with such properties can exist, including in a comment on our own post from an academic who’s written about the Riemann Hypothesis extensively. Dick Lipton and Ken Regan have written a blog post looking at the detail of how $T$ is supposed to behave. According to some sources, Atiyah has stated he’ll be publishing a more detailed paper shortly, but not many are holding their breath.

This is not easy mathematics, and even top-level mathematicians sometimes find their proofs don’t hold together; it’s no surprise that a solution to this problem wouldn’t be found in a few lines of mathematics, and it’s just a shame that this was so built up and sensationalised when it’s becoming obvious that Atiyah didn’t consult with anyone else about his proof before presenting it in a public forum.

Atiyah’s is by no means the first claimed proof of the Riemann Hypothesis of recent years; many end up in the wastepaper bins of academic mathematicians around the world, who get sent them in handfuls. Only some claimed proofs get this level of attention though, and this one has been somewhat unique in that it was taken a lot more seriously than usual.

You might remember that a few years ago, the BBC reported that Opeyemi Enoch had proved the Riemann hypothesis, and we immediately said “no, he hasn’t”. Now Sir Michael Atiyah claims he’s proved it, and before his lecture we said “well, let’s see”. Why the difference?

Enoch looked like any crank: went straight to the wrong venue – BBC News, not other mathematicians – didn’t publish his proof in the open, and presented it at a very scammy looking conference run by Nina Ringo, who has previous form. And we’d never heard of him: the false-positive rate for “anyone on the planet” with a claimed big proof is much, much higher than for, say, professors of maths in big universities. So for all those reasons, we were fairly sure he hadn’t done it, like I can fairly confidently say you’re not holding a winning lottery ticket right now.

Atiyah, on the other hand, has a Fields medal, an Abel prize, and a very long list of publications. We still thought he probably hadn’t proved it, based on his other recent pronouncements, but he said he was going to present his proof, so we gave him the benefit of the doubt.

It’s also valid to ask, why was Atiyah given such a prestigious platform to present his work? Well, the HLF isn’t an ordinary academic conference – on their own website, they stipulate that “the Heidelberg Laureate Forum is not a classical scientific conference but a networking event meant to motivate and inspire the next generation of scientists.” The Forum doesn’t vet the talks submitted by the Laureates, as up until now they’ve never needed to – most people give talks about their historical achievements, or occasionally the work of their students, and some even give the same talk every year. It’s still fair to say that a word of advice to Atiyah, to maybe stick to a general historical account of the problem (which his talk mostly was) might have been useful, if he could have been convinced.

Atiyah’s proof is a very good example of what a proof isn’t: it doesn’t contain enough detail to convince someone of the truth of his argument, or even enough detail for someone to point to as wrong. There simply wasn’t much to it, and the real glory of maths is in the gritty technical details, all of which were missing from this. Broad strokes are useful to give an overview, but they need to be backed up with working out.

The press has been patchy in its coverage of both Atiyah’s presentation and the reaction to it. It’s been a good example of how ill-equipped the general press is to cover maths news – breathless coverage from The Times, The Mirror and The Independent reveal a distinct lack of skepticism – but for anyone outside of mathematics, it’s not easy to know whether skepticism is warranted. At least some outlets looked like they’d tried to ask a professional mathematician, but many claimed that nobody was prepared to comment – a point that Jacob Aron of *New Scientist* has complained about.

Quite a few places, including this Science Alert article (which was shared by the Australian Mathematical Sciences Institute on Twitter!) make slightly inaccurate claims, including “the hypothesis has been checked for the first 1,000,000,000,000 prime numbers”, and that’s not really how that works. Fortunately, we didn’t see anyone saying “it’s definitely proved”, but did see a few stories framed as “rivals disagree”, which isn’t quite how this went down.

One positive side effect of this whole affair has been the amount of buzz it’s created around the Riemann Hypothesis, big results like this, and maths in general. It’s reminded us how exciting it would truly be if someone were to prove a result of this nature, and it was cool to see hundreds if not thousands of mathematicians (and non-mathematicians) talking about the proof on social media.

“Being wrong isn’t a bad thing like they teach you in school. It is an opportunity to learn something.” – Richard Feynman

Some commented ahead of the proof announcement that they hoped it was a real proof, as otherwise it would be terrible for someone like Atiyah, being publicly seen to be totally wrong about something. This sentiment, or equivalents post-hoc, have been echoed with varying levels of concern for his reputation. We think that’s a really unhelpful thing to say: being wrong is an inevitable part of mathematics research, and it happens to everyone. Promulgating the idea that maths is all about being 100% right all of the time not only makes it harder for people at research level but is also one of the major reasons people are put off maths at an early age. Yes, it is possible on occasion to come up with an unassailably correct solution to a maths problem, unlike other disciplines which are more subjective in their assessment, but terror of failure causes some people to freeze up and not even try – when in fact being wrong is inevitable, and an important part of the learning process. That said, there are less public forums in which to be wrong, and as we’ve discussed, steering Atiyah towards one of those might have been kinder.

We wish Sir Michael a pleasant retirement in the knowledge that he has made a huge contribution to maths over the course of his life, and that shouldn’t feel lessened or soured by these events. It’s fantastic that someone can continue working on mathematics into their old age, especially if, through collaboration, they can inspire the next generation. We are grateful to him for his work.

I loved the message you tried to convey through this article. Mathematics isn’t about being always right but giving a room for further discussions and inspiring young minds.

Irrespective of the outcome from this conference, there is a message for human endeavour and perseverance. Thanks for the balanced approach towards the event!

Keep Calm

&

Enjoy Maths

Two small points of disagreement (but in general thank you for your excellent coverage of this story). First, it has been suggested often, and you appear to repeat the suggestion, that if the Riemann hypothesis is proved, then it would threaten security on the internet. But unless I’m missing something (which is possible), that’s just plain wrong. If you write an algorithm that depends on the truth of the Riemann hypothesis, it doesn’t make any difference to the running of that algorithm whether we have a

proofof the truth of the Riemann hypothesis. As far as I can see, the idea that the Riemann hypothesis has some bearing on cryptography is based on a fantasy that if we could prove the Riemann hypothesis, we’d get such a huge insight into the behaviour of prime numbers that we’d suddenly become much better at splitting products of two large primes into their constituent factors. But there is no detail to back up this fantasy — the factorization problem is a very different problem from questions about the distribution of prime numbers.The second point where I disagree concerns your statement that being wrong is a normal part of mathematics. Of course, that statement itself is 100% true, but what you suggest by making it is that Atiyah came up with an argument that had the kind of mistake that experienced mathematicians often make, whereas that is not at all an accurate description of his paper. (It would have been an accurate description of the paper by Ed Nelson a few years ago that claimed to prove that ZF was inconsistent. That paper did contain a subtle mistake, and Nelson retracted his claim the moment the mistake was pointed out.)

I agree that we should not be afraid to make mistakes. One can justify that on efficiency grounds: if you don’t make 100% certain that everything you do is correct straight away, you can work so much faster that even if only a certain fraction of your arguments turn out to be correct when you try to fill in the details, you’ll probably produce a greater quantity of correct interesting mathematics. But that efficiency justification doesn’t apply to arguments that aren’t real arguments at all and quite clearly don’t have a chance of working.

Having said that, I wholeheartedly agree with you that this episode doesn’t have the slightest effect on Atiyah’s amazing legacy to mathematics. And I also agree that it has led to some interesting mathematical discussions.

I don’t think your description does justice to the relation between RH and Internet security. True, the RH does not give us an algorithm to break RSA or its elliptic curve variants. The encrypted data is not immediately compromised. But there are a small handful of conjectures people take into account when designing these cryptosystems, and these conjectures do tend to rely in turn on RH. If RH turned out to be wrong, we would no longer have confidence in cryptosystems like RSA that assume, for example, that factoring large numbers is hard.

We would also have problems with key generation, which relies on RH to know that the methods used really do come up with prime numbers and not just very strong pseudo-primes.

So the press got it backwards: if Dr. Atiyah had proved RH is false, that would be the threat to our cryptosystems. But since most of them assume it is true anyway, proving it does not have an impact on them.

@Matthew Johnson:

The RSA cryptosystem requires finding large random primes which one typically generates by (a) picking a large random number and (b) running a primality test such as the one of Agrawal, Kayal and Saxena (see https://en.wikipedia.org/wiki/AKS_primality_test) to determine whether this number is indeed prime. If (b) fails, try again. This procedure finds a prime quickly (that is, in not too many repetitions) since PNT tells us that primes are pretty dense in the integers. RH “merely” gives us a prediction on the right asymptotic behavior of the (low order) error term in PNT and has little relevance to how quickly one finds prime numbers. Therefore, I do not see how the following statement is justified: “We would also have problems with key generation, which relies on RH to know that the methods used really do come up with prime numbers and not just very strong pseudo-primes.”

Re: “If RH turned out to be wrong, we would no longer have confidence in cryptosystems like RSA that assume, for example, that factoring large numbers is hard.” I don’t see any relation between efficient factoring algorithms and the truth of RH. If you know of any result along these lines, I’d be curious to know.

He shouldn’t really claim he’s proved it if he can’t back it up. This isn’t a mathematical error, its a philosophical error. The commonness of errors in mathematical enquiry are quite irrelevant here.

Aside from the reasons that Sir Michael Atiyah might make this kind of erroneous claim, people make sincere claims of proofs that later turn out to be wrong all the time. We’ve covered a few of the more high-profile ones here in the past.

Thanks for your comment. Yes, we could’ve made it clearer that Atiyah’s proof being wrong isn’t the only thing keeping public key encryption safe.

I also agree that the kind of error Atiyah has made isn’t the kind of error you’d expect to make in normal circumstances.

Thanks. Yes, the comment about encryption was meant as slightly tongue-in-cheek, if not adequately flagged as such (Paul wrote a little about how such claims get vastly exaggerated in our post last Sunday) and it’s frustrating how many news outlets have been pushing that angle.

I think your distinction between two types of wrongness is accurate, but it’s again a subtlety that many readers of the coverage this gets won’t be able to really grasp, and it’d be a shame if people’s main takeaway from this whole story is that he’s wrong and that’s all that matters. (I think I’m also trying to be as generous as possible in my interpretation of Atiyah’s motives, as none of us really know what’s going on with him personally and as far as I can tell he’s at least being sincere.)

The connection between RH and primes is this: the prime number theorem gives an estimate on the number of primes less than n, and the current bound on the error of that estimate is exponential in n (so the estimate is essentially useless). If RH is true, the error is bounded by sqrt(n). This means factoring algorithms can know (almost) the exact number of primes less than the number they are factoring. That may not be 100% useful, but I think it will provide some benefit (a speedup in current algorithms at least).

You right, we know the Riemann hypothesis works and therefore can zeta function can be used by cryptographers. However, its proof has theoretical importance.

Timothy Gowers said: “As far as I can see, the idea that the Riemann hypothesis has some bearing on cryptography is based on a fantasy that if we could prove the Riemann hypothesis, we’d get such a huge insight into the behaviour of prime numbers that we’d suddenly become much better at splitting products of two large primes into their constituent factors. But there is no detail to back up this fantasy — the factorization problem is a very different problem from questions about the distribution of prime numbers.”

But do you agree that we cannot exclude the fantasy altogether? The Riemann hypothesis is so difficult that the person who eventually proves it (or disproves it!) will probably need to invent whole new mathematical concepts, idioms, functions. We are only as good as the tools we have. If we want to go further, we must build new tools. That is what will be interesting about the proof. Not the fact of the proof, but the ingenuity that it is likely to contain. Someone should write a thriller about that – it will make a great movie ;)

Clear explanation on the article.I hope for a different approach in solving the Riemann hypothesis… Whether true or false, a new kind of approach likened to someone like Newton thinking why an apple fell down instead of upwards.

When I first watched a presentation, it struck me that it was more like one of those youtube videos explaining the proof of Fermat’s last theorem (not much detail, a lot of high-profile mathematical terminology, and zero explanations). It seems that what sir Atiyah gave was a sketch of a possible technique of a proof, and not a proof itself.

About this line in the text “becoming obvious that Atiyah didn’t consult with anyone else about his proof before presenting it in a public forum.” That’s not true. I have evidence (copies of the emails) that Atiyah tried to discuss his idea with many prestigious mathematicians. I do not know whether they replied him or not (neither I know whether he followed some advice), but at least I know he tried to communicate with others. Did you try to ask Atiyah about this point?

To clarify – no, we haven’t spoken directly to Atiyah about this, but the intention was to say: it’s becoming apparent there wasn’t any fruitful discussion about it. The proof is missing a lot of detail, and it needed a lot more work (or fundamentally rethinking) before it should have been presented, and from what we’ve seen, anyone he spoke to could have told him this. If he did try to contact people, they either didn’t reply or he ignored their advice, both of which are a shame.

As someone who simply has an interest in math, the buzz generated on this topic has gotten me to look into the Riemann Hypothesis. I’ve only heard of it before. Now I have a much greater understanding of the Reimann zeta function and the implications of the hypothesis.

I thank What the idea of Atiyah to use the fine structure constant with the Todd function is Vera interesting,the way Must to be corrrect.the use for spontaneous symmetry breaking be other tools usei in the RH.maybe the RH be something irreal

I think the poor reporting in the mainstream media is the result of non-mathmaticians not realising that all the mathematicians expressing scepticism or saying the proof is interesting but needs a little more work were just being polite. While I can certainly understand the desire to protect a man that has made an enormous contribution to mathematics from public humiliation, it has been quite harmful to the public understanding of mathematics. I’ve seen a lot of people that are waiting for the results of peer review, not realising that pretty much every mathematician on the planet has already reviewed it and privately reached the same conclusion and there isn’t going to be a formal peer review process because there is nothing to formally review. It is not good that the mathematical community has created that kind of confusion through its attempts to be kind to an old man.

I have offered my mathematician friends $10,000, evens, that no proof of RH will emerge from Atiyah’s delusional scribblings within 5 years of his Heidelberg address.

Only the more foolish of them showed any interest.

Further, a proof of RH is highly unlikely to impact adversely on cryptography.

F.C.V. Hardy-Head (an anagram)

feel this is all accurate but also kind of misses the point, it lacks context. Atiyah has cried wolf 3 times in the last year with Feit-thompson proof, 6-spheres, and now Riemann. theres also wild physics claims about the fine structure constant. these are all extraordinary problems and to solve a single one would be revolutionary. all have been basically rejected by reviewers according to internet rumors although some posted by insiders. the prior proofs are circulating somewhat privately so its understandable that there is not much knowledge of these other cases. again according to some (anonymous) reddit comments he has been told face-to-face of incorrect aspects of the papers by top other mathematicians. many are choosing not to say anything publicly so that Atiyah can “save face” or whatever. one has to look at the bigger picture and one is kind of forced to suspect this is more than merely “eccentricity” going on here at this point. do wish Atiyah the best but hes got to listen to his own peers better, maybe work on 1 revolutionary proof at a time, and officially retract any that he realizes are incorrect, and some lost respect can be regained. much more detail here https://vzn1.wordpress.com/2018/09/26/atiyah-riemann-attack-post-mortem-autopsy-primes-torch-carried-on/

A wonderful side effect of Mr Atiyah’s efforts to prove RH is that all around the world many people is trying to understand a little bit about it. Young minds had been led to a beautiful mathematical garden such as Theory of Numbers.

> A wonderful side effect of Mr Atiyah’s efforts to prove RH is that all around the world many people is trying to understand a little bit about it.

I hope you are either young or joking.

Atiyah has, sadly, lost his marbles or his judgement. He now has form. His “proof” contains no proof that a fn (“T”) with the properties he specifies exists. “FIRST PROVE EXISTENCE!” For a variety of reasons that cannot fit into this margin, I further believe that no such T can exist, shutting the door to this method of attack on RH.

And, associating mathematics with nonsense and absurd claims has no positive effect and, in fact, fuels public distrust in logic and mathematics.

On to the crank.

Personally, “Dr” Opeyemi Enoch should be dismissed from his government position for deliberately bringing his country into (further) disrepute, if indeed that is possible in the case of Nigeria. Public flogging and being stripped of his academic qualification (by whatever fifth-rate institute granted it) would be acceptable add-ons. Alternatively, was he crazy/stupid enough to think either that his unlearned scribblings had proved RH, or that the Clay Institute were going to hand him the $1m without careful peer review?

I think public flogging is a bit severe!

Between vitriolically attacking Atiyah for what, I would point out, is a MATHS issue, not any kind of misdeed, and your equally vitriolic and even possibly racist rant about Enoch, you’re not coming off very well.

If you’re a professor and a professional mathematician, you should be able to respectfully criticize your colleagues’ work. Flying off the handle like this over such a trivial thing is much worse than producing a vague “proof” of the Riemann hypothesis.

Atiyah’s error seems to lie in the belief that he cannot make an error. That arrogance is on full display in his presentation, when he suggests that ageism is the only possible cause of his proof not being published. There are two sides to prejudice; both real and imagined prejudice are damageous. The whole saga is a fascinating insight into the prejudicial workings of the Scientific community. Your article is biased by Atiyah’s past and so, you’re incapable of not putting a generous spin on an incredibly embarrassing display. If past success can so distort reality, one has to wonder how many similar distortions have successfully slipped through the cracks and allowed Scientific illusion to become Scientific fact. My favorite example of this is Feynman’s fantastical clouds of probability explanation in quantum physics. That single fairytale has voided so much potential progress. We are now rapidly approaching the limits of what can be known in math and physics – it seems the broader community could be more well organized around the problem of vetting proofs and theories. This observation is tangential to the point you make well about being brave enough to be wrong.

Really, really appreciated this article. I’d just read the ScienceAlert article that you’d mentioned and decided to do a bit of deeper digging into the topic. If anyone can answer though, what is the relation between the zeta function and prime numbers? What I have read has said the distribution of zeta function solutions and prime numbers is “similar” or “closely related”, but wouldn’t a proof’s application to prime numbers be useless unless there’s an identical match?

Although I understand the reasons for taking a balanced view and make an empathic conclusion from a social point of view, I think one should hold the same standards for everyone working in an exact science. If one would take Atiyah’s presentation and judge it just by its content, and not by the history associated with the author, there seems to be hardly any positive thing to say about the proof part. Such a ‘proof’ coming from any other author would have been instantly, and possibly harshly, discredited. Thus, it is just a plain insult on proper scientific work and gives a very bad example for any aspiring young scientist to see that previous merits, whatever good they are, can create a bias in such a way. If it comes to the search for truth, there should be no exceptions for no one.

Beautifully written.

Absolutely agree with this statement. Wishing Sir Michael nothing but the best. :)