# Riemann hypothesis possibly solved:

sonhouse
Science 27 Sep '18 01:34
1. sonhouse
Fast and Curious
27 Sep '18 01:34

Still to be verified but he already has fields medal and Abel Prize.
2. 27 Sep '18 04:07
Originally posted by @sonhouse

Still to be verified but he already has fields medal and Abel Prize.
Let's wait for the verification. Then we cheer!
Remember the Fermat Last Theorem? They cheered too early.
3. sonhouse
Fast and Curious
27 Sep '18 11:36
Originally posted by @fabianfnas
Let's wait for the verification. Then we cheer!
Remember the Fermat Last Theorem? They cheered too early.
True, but they did cheer after they worked it over.
4. 27 Sep '18 12:08
Originally posted by @sonhouse
True, but they did cheer after they worked it over.
But dr Wiles had some sweaty times before that...
He didn\t know wether or not his proof was flawed or not.
5. sonhouse
Fast and Curious
27 Sep '18 20:091 edit
Originally posted by @fabianfnas
But dr Wiles had some sweaty times before that...
He didn\t know wether or not his proof was flawed or not.
No but independent mathematicians found the disconnect which Dr. Wiles connected and then the proof was real.

Obviously this vetting of Riemann will take awhile.

Do you have any idea of what it will mean for other branches of math if this is proven true?

Maybe some connection to cryptography.
6. 28 Sep '18 06:25
Originally posted by @sonhouse
No but independent mathematicians found the disconnect which Dr. Wiles connected and then the proof was real.

Obviously this vetting of Riemann will take awhile.

Do you have any idea of what it will mean for other branches of math if this is proven true?

Maybe some connection to cryptography.
We consider the Riemann true already. Every proof needing Riemann should today be labelled a hypothesis. Riemann has, up to today, been used as an axiom, and we cross our fingers that this axiom is correct.

Let's wait for the day when the proof is considered correct.
7. 28 Sep '18 11:46
Originally posted by @fabianfnas
But dr Wiles had some sweaty times before that...
He didn\t know wether or not his proof was flawed or not.
And indeed, his first proof was flawed. The cheering only started properly after he corrected his proof.
8. sonhouse
Fast and Curious
28 Sep '18 13:571 edit
Originally posted by @shallow-blue
And indeed, his first proof was flawed. The cheering only started properly after he corrected his proof.
Of course. Proof's have been ventured before but flawed and proven so. So we will hold the champagne till it is fully vetted.

So Fnass says the math world has been assuming Riemann was true and going with that. So I guess not much will change except in the theoretical math world.

Here is an answer from Mathematics Stack Exchange:

"The Millennium problems are not necessarily problems whose solution will lead to curing cancer. These are problems in mathematics and were chosen for their importance in mathematics rather for their potential in applications.

There are plenty of important open problems in mathematics, and the Clay Institute had to narrow it down to seven. Whatever the reasons may be, it is clear such a short list is incomplete and does not claim to be a comprehensive list of the most important problems to solve. However, each of the problems solved is extremely central, important, interesting, and hard. Some of these problems have direct consequences, for instance the Riemann hypothesis. There are many (many many) theorems in number theory that go like "if the Riemann hypothesis is true, then blah blah", so knowing it is true will immediately validate the consequences in these theorems as true. A solution to some of the other problems is (highly likely) not going to lead to anything dramatic. For instance, the P vs. NP problem. I doubt anybody in the world thinks it is even remotely slightly conceivable that P=NP. The reason it's and important question is not because we don't (philosophically) already know the answer, but rather that we don't have a bloody clue how to prove it. It means that there are fundamental issues in computability (which is a hell of an important subject these days) that we just don't understand. Solving P≠NP will be important not for the result but for the techniques that will be used. (Of course, in the unlikely event that P=NP, enormous consequences will follow. But that is about as likely as it is that the Hitchhiker's Guide to the Galaxy is based on true events.)

The Poincare conjecture is an extremely basic problem about three dimensional space. I think three dimensional space is very important, so if we can't answer a very fundamental question about it, then we don't understand it well. I'm not an expert on Perelman's solution, nor the field to which it belongs, so I can't tell what consequences his techniques have for better understanding three dimensional space, but I'm sure there are."

https://math.stackexchange.com/questions/404624/what-does-proving-the-riemann-hypothesis-accomplish
9. lemondrop
pawn grabber
29 Sep '18 21:211 edit
can anyone explain to me how to compute the non trivial zeros of the beta function?
is it difficult?

don't know if I'm asking properly or making sense
10. HandyAndy
Non sum qualis eram
29 Sep '18 21:29
Originally posted by @lemondrop
can anyone explain to me how to compute the non trivial zeros of the beta function?
is it difficult?

don't know if I'm asking properly or making sense

https://qntm.org/riemann
11. 30 Sep '18 05:541 edit
Originally posted by @sonhouse

Still to be verified but he already has fields medal and Abel Prize.
I saw the question "Did the Riemann hypothesis get solved?"
And the answer from Luboš Motl, former Harvard Junior Fellow went:

<quote>
"No, it didn’t. I was excited while expecting the proof because I think that Atiyah’s life-long exceptional mind as well as his specialization within mathematics (especially things like the Atiyah-Singer theorem that could apply to “solutions” of an operator equation corresponding to the zeroes) made him a wonderful candidate to be able to solve the problem. He seems full of energy and he answered my e-mail within 5 minutes on Friday.

But as I discuss in Nice try but I am now 99% confident that Atiyah's proof of RH is wrong, hopeless, his proof doesn’t work at all and the talk – and especially the papers – are full of statements that the experts know to be incorrect.

He has used a special proposed function of the complex variable, the Todd function T(s), which is polynomial in convex regions of the complex plane but non-polynomial elsewhere. Already at this point, I am rather convinced that no such natural function exists. If something is a polynomial in a region, it is a polynomial everywhere. There is no natural canonical extension than the straightforward “analytic continuation”. So I believe that his concept of “weakly analytic” functions is used incorrectly as he sloppily jumps between analytic functions of real and complex variables – which differ in some important aspects.

More importantly, his proof doesn’t seem to use the special character of the zeta function – and its relationship with the sum or Euler formulae or with the distribution of primes – at all. Like 90% of hopeless “proofs” of RH, it seems to use the symmetry properties of zeta(s) in the complex plane only, and claim that out-of-the-critical-axis poles cannot exist. But one may easily modify the zeta function by removing some “well-behaved” zeroes on the critical axis while adding quadruplets of “wrong zeroes”. Atiyah’s proof seems to apply to this “modified” zeta function as well – and claim that it can’t have off-the-critical-axis poles. But by construction, it has such poles.

There are lots of other thought-provoking but deeply misguided claims in the papers. He claims to derive the fine-structure constant in the physics of electromagnetism as if it were similar to “pi” – a renormalized “pi”. Lots of numerologists love to propose some purely mathematical, simple formulae for the fine structure constant which is about 1/137.035999. But the fine-structure constant is a complicated beast that finely depends on properties of all elementary particles, including three generations of quarks and leptons as well as the Higgs boson, and both in quantum field theory and string theory, it’s almost certainly vastly less unique than “pi”. There are many (in the stringy landscape picture, googol-like many) cousins of the Standard Model and each of them has dozens or hundreds of parameters similar to the fine-structure constant. None of them may be assumed to be a towering mathematical constant that is on par with “pi”. And even if this equality between the fine-structure constant and a “simple mathematical constant” held by some miracle, Atiyah hasn’t presented valid evidence backing this miraculous coincidence.

He also makes wrong claims that his proof wouldn’t be allowed in the ZF axiomatic system of set theory. If the complex-analysis proof worked, it would be totally independent of all ZF/GB specific axioms and of the Axiom of Choice. The independence follows from the logical simplicity and down-to-Earth character of the science about the complex functions.

Also, even if the Todd function T(s) existed with the properties that Atiyah promotes and needs, I believe that his proof by contradiction still wouldn’t work. The proof by contradiction starts by assuming the “wrong root” with the smallest imaginary part, and proving that there exists another one with an even smaller one – and that’s a contradiction (if one is the smallest one, there can’t be a smaller one). That’s what he says after the 15-line proof. This is a possible strategy (I actually guessed this strategy correctly in another Quora answer I wrote on Sunday) but there’s no known way to fill the details and reach the contradiction. But his 15-line-long proof doesn’t actually seem to use this strategy at all. Instead, it claims that a function F(s) constructed by composing zeta(s) and T(s) in some way has to be zero, and therefore zeta(s) has to be zero, too. I think that the steps leading towards the claim that F(s)=0 would be algebraically incorrect even if all the assumptions were right.

The papers are thought-provoking and you can see that he has thought about many deep parts of mathematics that are close to mathematical or theoretical physics and he still remembers lots of traces of this vast knowledge but the ways how he combines them these days are completely wrong and irrational, plagued by elementary mistakes in reasoning and algebra and dominated by wishful thinking.

I spent much of Monday with that event and it’s likely that I wouldn’t do it again if Michael Atiyah made a similar big claim. But I still see him as a role model – and if I were offered to be alive at age of 89+ and be this excited about the thinking about mathematics, I would accept the deal."
</quote>
12. sonhouse
Fast and Curious
30 Sep '18 16:45
Originally posted by @fabianfnas
I saw the question "[b]Did the Riemann hypothesis get solved?"
And the answer from Luboš Motl, former Harvard Junior Fellow went:

<quote>
"No, it didn’t. I was excited while expecting the proof because I think that Atiyah’s life-long exceptional mind as well as his specialization within mathematics (especially things like the Atiyah-Singer theor ...[text shortened]... 9+ and be this excited about the thinking about mathematics, I would accept the deal."
</quote>[/b]
Uh oh, trouble in river city. Well, we'll see what happens when the big boys get involved.
13. 30 Sep '18 18:01
Originally posted by @fabianfnas
We consider the Riemann true already. Every proof needing Riemann should today be labelled a hypothesis. Riemann has, up to today, been used as an axiom, and we cross our fingers that this axiom is correct.

Let's wait for the day when the proof is considered correct.
Well, if it isn't used as an axiom, we pretty much have to go back to square one with prime numbers.
14. 01 Oct '18 04:27
Originally posted by @ashiitaka
Well, if it isn't used as an axiom, we pretty much have to go back to square one with prime numbers.
Well, we use the Riemann as it was a true and proved theorem. Yet it isn't.
We cross our fingers and pretend it is true and proved, and continue to use it.

If it turns out to be false - how much of mathematics has to be redone?
15. 02 Oct '18 00:502 edits
Originally posted by @sonhouse

Still to be verified but he already has fields medal and Abel Prize.
For personal reasons, I am quite partial to Michael Atiyah, but even a genius can be human.

https://www.sciencemag.org/news/2018/09/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis

"Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis"

"But many experts doubt its validity, especially because Atiyah, 89, has been making mistakes in recent years."

"But despite a long and prolific career, the Riemann claim follows on the heels of more recent, failed proofs.
In 2017, Atiyah told The Times of London that he had converted the 255-page Feit-Thompson theorem,
a half-century-old theory dealing with mathematical objects called groups, into a vastly simplified 12-page proof.
He sent his proof to 15 experts in the field and was met with skepticism or silence, and
the proof was never printed in a journal. A year earlier, Atiyah claimed to have solved a
famous problem in differential geometry in a paper he posted on the preprint repository ArXiv,
but peers soon pointed out inaccuracies in his approach and the proof was never formally published.

Science contacted several of Atiyah’s colleagues. They all expressed concern about his
desire to come out of retirement to present proofs based on shaky associations and
said it was unlikely that his proof of the Riemann hypothesis would be successful.
But none wanted to publicly criticize their mentor or colleague for fear of jeopardizing the relationship."