Riemann hypothesis, the fine structure constant, and the Todd function

This morning Sir Michael Atiyah gave a presentation at the Heidelberg Laureate Forum with a claimed proof of the Riemann hypothesis. The Riemann hypothesis (RH) is the most famous open problem in mathematics, and yet Atiyah claims to have a simple proof.

Photo via https://twitter.com/QuinteScience/status/1044134956996468736

Simple proofs of famous conjectures

If anyone else claimed a simple proof of RH they’d immediately be dismissed as a crank. In fact, many people have sent me simple proofs of RH just in the last few days in response to my blog post, and I imagine they’re all cranks [1]. But Atiyah is not a crank. He won the Fields Medal in 1966 and the Abel prize in 2004. In other words, he was in the top echelon of mathematicians 50 years ago and has kept going from there. There has been speculation that although Atiyah is not a crank, he has gotten less careful with age. (He’s 89 years old.)

QuinteScience, source of the image above, quoted Atiyah as saying

Solve the Riemann hypothesis and you’ll become famous. But if you’re already famous, you run the risk of becoming infamous.

If Atiyah had a simple self-contained proof of RH that would be too much to believe. Famous conjectures that have been open for 150 years don’t have simple self-contained proofs. It’s logically possible, but practically speaking it’s safe to assume that the space of possible simple proofs has been very thoroughly explored by now.

But Atiyah’s claimed proof is not self-contained. It’s really a corollary, though I haven’t seen anyone else calling it that. He is claiming that a proof of RH follows easily from his work on the Todd function, which hasn’t been published. If his proof is correct, the hard work is elsewhere.

Andrew Wiles’ proof of Fermat’s last theorem was also a corollary. He proved a special case of the Taniyama–Shimura conjecture, and at end of a series of lectures noted, almost as an afterthought, that his work implied a proof to Fermat’s last theorem. Experts realized this was where he was going before he said it. Atiyah has chosen the opposite approach, presenting his corollary first.

Connections with physics

Atiyah has spoken about connections between mathematics and physics for years. Maybe he was alluding to his work on the the fine structure constant which he claims yields RH as a corollary. And he is not the only person talking about connections between the Riemann hypothesis specifically and physics. For example, there was a paper in Physical Review Letters last year by Bender, Brody, and Müller stating a possible connection. I don’t know whether this is related to Atiyah’s work.

Fine structure constant

The fine structure constant is a dimensionless physical constant α, given by

\alpha = \frac{e^2}{4 \pi \varepsilon_0 \hbar c}

where e is the elementary charge, ε0 is vacuum permittivity, ħ is the reduced Planck constant, and c is the speed of light in a vacuum. Its value is roughly 1/137.

The Todd function

The Todd function T is a function introduced by Atiyah, named after his teacher J. A. Todd. We don’t know much about this function, except that it is key to Atiyah’s proof. Atiyah says the details are to be found in his manuscript The Fine Structure Constant which has been submitted to the Proceedings of the Royal Society.

Atiyah says that his manuscript shows that on the critical line of the Riemann zeta function, the line with real part 1/2, the Todd function has a limit ж and that the fine structure constant α is exactly 1/ж. That is,

limy → ∞ T(1/2 + yi) = ж = 1/α.

Now I don’t know what he means by proving that a physical constant has an exact mathematical value; the fine structure constant is something that is empirically measured. Perhaps he means that in some mathematical model of physics, the fine structure constant has a precise mathematical value, and that value is the limit of his Todd function.

Or maybe it’s something like Koide’s coincidence where a mathematical constant is within the error tolerance of a physical constant, an interesting but not necessarily important observation.

Taking risks

Michael Atiyah is taking a big risk. I’ve seen lots of criticism of Atiyah online. As far as I know, none of the critics have a Fields Medal or Abel Prize in their closet.

Atiyah’s proof is probably wrong, just because proofs of big theorems are usually wrong. Andrew Wiles’ proof of Fermat’s Last Theorem had a flaw that took a year to patch. We don’t know who Atiyah has shown his work to. If he hasn’t shown it to anyone, then it is almost certainly flawed: nobody does flawless work alone. Maybe his proof has a patchable flaw. Maybe it is flawed beyond repair, but contains interesting ideas worth pursuing further.

The worst case scenario is that Atiyah’s work on the fine structure constant and the Todd function is full of holes. He has made other big claims in the last few years that didn’t work out. Some say he should quit doing mathematics because he has made big mistakes.

I’ve made big mistakes too, and I’m not quitting. I make mistakes doing far less ambitious work than trying to prove the Riemann hypothesis. I doubt I’ll ever produce anything as deep as a plausible but flawed proof of the Riemann hypothesis.

Update

The longer paper has been leaked, presumably without permission from Atiyah or the Royal Society, and it doesn’t seem to hold up. No one is saying the proof can be patched, but there has been some discussion about whether the Todd trick could be useful.

In writing this post I wanted to encourage people to give Atiyah a chance, to wait until more was known before assuming the proof wasn’t good. I respect Atiyah as a mathematician and as a person—I read some of his work in college and I’ve had the privilege of meeting him on a couple occasions—and I hoped that he had a proof even though I was skeptical. I think no less of him for attempting a to prove a big theorem. I hope that I’m swinging for the fences in my ninth decade.

Related posts

[1] I don’t call someone a crank just because they’re wrong. My idea of a crank is someone without experience in an area, who thinks he has found a simple solution to a famous problem, and who believes there is a conspiracy to suppress his work. Cranks are not just wrong, they can’t conceive that they might be wrong.

20 thoughts on “Riemann hypothesis, the fine structure constant, and the Todd function

  1. Pablo Sebastian Armas

    Sorry for this,

    I don’t mean to be disrespectful to anyone. I do not understand in what part of the proof any property of the zeta function is used.

    Pablo

  2. Pablo, I understand. The zeta function is not explicit in his HLF talk. And somewhere the proof of a conjecture about the zeta function has to mention the zeta function! I don’t know where the connection is, but I haven’t read the fine structure paper.

  3. It’s not in the fine structure paper. This is not a “plausible but flawed attempt” because it is not plausible.
    Atiyah has showed the manuscript to other well-regarded mathematicians, who told him not to publish or give talks about it. He told them they were wrong. Just like the paper on complex structures on the 6-sphere. His paper was rejected from the arXiv, and he believes that this is because of ageism. Atiyah’s exceptional work has been on differential geometry and algebraic topology; he has done no previous work in analytic number theory.

    Unfortunately, this fits all of your criteria for crankery.

  4. I’m sorry to hear that. I was starting to fear that might be the case based on the conspicuous silence from experts. Their silence says more (and is more respectful) than the Twitter snipers criticisms.

  5. The fine structure constant is defined as a ratio involving 4 fundamental physical quantities. So I guess what it would mean for it to have a precisely definable mathematical value is that there are really only three fundamental physical quantities here.

  6. The best comment / critique I have read on Atiyah’s purported proof. Judicious, prudent, and (unlike many others) respectful to the great Atiyah

  7. Is the fine structure constant directly measurable? I believe that the constituent quantities like c e and hbar are measured and the uncertainties propagated. Since the fine structure is dimensionless people have held out hope that it might pop out of a convergent series and give clues towards a deeper version of QFT; for instance this is given as a hw problem in Griffith’s QM book (with the warning to not spend too much time on it). An added benefit of such an expression would be that it could lower the uncertainty of the most uncertain constituent of the expression. It doesn’t appear that Atiyah has succeeded, but this form of numerology is indeed popular and potentially useful.

  8. Hello,
    the f.s constant is a dimension less constant it is a pure mathematical number, it is not related to units like meter or second or joule. He could be right about it.

  9. I don’t understand this method and maybe somebody can help me with this.
    As I understand, the proof is about applying the Todd function to the zeta function. The question is, If you apply the Todd function to the prime zeta function or even any Dirichlet series then you also can prove that any of those functions have no zeros in the right half critical strip. Right?! The problem is that those functions have zeros on that strip.

  10. The problem is that (so far), there are no known description of the Todd function in details , it could also be a lucky coincidence, the F.s. constant is known to only 10 digits. The exact value is not known because the speed of light has been fixed to be a certain number (299 792 458 meters per second) by decree. This means that after the 10’th digit is pure speculation.
    But at the same time, Mr Atiyah is not exactly an amateur.
    The Todd function , from what I have read is only similar to the Zeta function and certainly does not behave as the Zeta with zeroes on the critical strip. If you look in the standard litterature about the R.H. there are many equivalent formulations. If the breakthrough of Mr Atiyah is true then there are many consequences in regards physics as well as mathematics, I would be very interested to read the article on the connection of the Zeta function, the Todd function and the F.S. constant.
    If true , in my opinion it is a major event, connecting number theory to physics directly .

  11. Isn’t in a real sense an empirical physical constant? We might view it differently now, but when it was first discovered, it was very much just something empirical. And it seems to be based on the real geometry of our universe; other universes (with different geometries/curvature) could have different values for .

  12. It’s a ratio of empirically measured constants. It’s possible that these constants are not independent, that there’s one less degree of freedom than is apparently the case. It’s also possible that this is just a coincidence. The details are sketchy to say the least.

  13. Looking very briefly at this, it seems to me the critical step is that when applying the Todd function to the Zeta function to get the F function, Atiya claims that within the critical strip the new function should have the property F(2s) = 2F(s) because of convexity of the region, so this should require some properties of the Todd and Zeta functions within convex regions. This may also apply to other functions than the zeta function when composed with the Todd function…

    He also says because F(s) is analytic at zero (a property of the Todd function and the Zeta function composed) that together with the above property, if it’s zero at another point in this strip (off the line) then it must be zero at all points in the strip. I don’t quite follow this. There are nonzero analytic functions with more than one zero, but I’m guessing within a strip they don’t have the F(2s) = 2F(s) property. It’s been too long since I thought about complex analysis to know whether those two properties imply identically zero.

    The real question is what is the special property of the Todd function that allows you to compose it with the Zeta function and get a new function with certain properties above? This is where special properties of the zeta function come in (or perhaps more likely a class of functions of which the zeta function is a member).

  14. Is there at least some computational evidence that the Todd function approaches (or at least seems to approach) 1/alpha? If so, this may be interesting in itself even if the RH proof turns out to be fatally flawed.

  15. “Andrew Wiles’ proof of Fermat’s last theorem was also a corollary. He proved a special case of the Taniyama–Shimura conjecture, and at end of a series of lectures noted, almost as an afterthought, that his work implied a proof to Fermat’s last theorem. ”
    I don’t think that is a fair statement. The fact that Taniyama-Weil implied FLT was the result of a lot of fantastic work by a number of mathematicians starting with Frey’s insight that FLT was connected to elliptic curves and including major contributions by Serre and Ribet. Just my opinion.

  16. The idea of ​​Professor Atiyah is very seductive. If pi and e are immersed in nature why not the fine structure constant?

    However, I do not understand the relationship with the Riemann zeta function until now. But that’s my fault.

  17. I would like to suggest to Atiyah the following: T(Pi)=1/alpha is not a relationship between pi and alpha in reality but instead a relationship between pi and h (the Plank constant) ! I have a theory that argues that pi and h are holographic to each other. This Todd formula from Atiyah is exactly saying the same. It is the tip of an enormous iceberg in fact.

  18. Professor Atiyah does mention in his talk that he was not trying to solve RH but rather working to understand the fine structure constant.

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