α, π, and the Riemann Hypothesis

On September 24th an accomplished mathematician named Sir Michael Atiyah gave a presentation wherein he claimed to have discovered a simple proof of the Riemann Hypothesis, a 160-year-old open question in mathematics about the distribution of prime numbers. Most people making such a claim would be immediately dismissed (longstanding open questions like that in mathematics are usually solved as the result of many people working together rather than a single person), but Atiyah has won both the Fields Medal and Abell prize (both considered roughly the equivalent of a Nobel Prize for mathematicians) and has enough credibility to cause people to take notice.

While this is interesting enough on its own for its many profound implications across mathematics, it's extremely interesting (and personal) for me due to the fact that Atiyah's claimed proof of the Riemann Hypothesis was apparently a happy accidental byproduct (!) of his true goal: finding a way to compute the value of the fine-structure constant, α based on other numbers. If you don't know, my entire Ph.D. revolves around searching for variation in α; if this claimed proof were to turn out to be true (and I'll revisit that if in a second), it would elevate α to the same level as e or π, an unchanging mathematical quantity. As far as I can tell, his calculation of the value of α would have it be proveably constant, which would put paid to the notion of searching for variation in it, and incidentally my entire Ph.D..

Now, I'm not getting too worried about this just yet for a few reasons. First of all, multiple people who know far more than I do about the relevant mathematics have expressed skepticism about the results. Atiyah, though undoubtedly incredibly smart and gifted, is getting on in years (he's 89), and has advanced a few theories in the past couple of years that have failed to gather peer support. It seems very unlikely that such a longstanding open question has a simple proof that no one has spotted until now. It's not impossible, especially as advances are made in mathematics over time, but, while romantic, the idea of a lone genius stumbling upon a profound proof is less and less likely nowadays where significant advances are increasingly the result of collaboration and correspondence between large teams of people.

Now, I'm only a humble physicist and no mathematician, and will freely admit that I don't understand probably the majority of the math behind the claimed proof, but I have three things that make me skeptical myself.

There's just something about α that seems to attract numerological explanations. Being a dimensionless physical constant with no known relation to other important mathematical constants or way of calculating its value seems to fire people's imaginations. Richard Feynmann in 1985 wrote of α:

Immediately you would like to know where this number for a coupling [α] comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil."

Soon after α was introduced by Arnold Sommerfeld in 1916 people began coming up with schemes for how it relates to various mathematical constants. Atiyah's proof claims a sort of connection with π, with 1/α being the limit of some kind of “renormalization” function acting on π. Numerological explanations have come and gone over the last century with none of them ultimately being accepted; while it's possible this is an exception, it's certainly not the first attempt someone has made to derive α from other numbers.

However, this leads to a second consideration: α is a measured quantity. We don't have a method to calculate its value now (that's the whole reason behind my Ph.D.), and it's difficult to see how to prove that, even if the particular function Atiyah has introduced (called the Todd function) works as he claims, that it's actually producing the real, correct value of α and isn't merely a coincidence.

Finally, I'm a τ-ist; I believe that the correct circle constant is \(τ=2π\), so I find it unlikely that an explicit mathematical connection would exist between π and α. It's not impossible, certainly, but I strongly suspect that if such a connection exists it would be between α and τ, not π. Interestingly, I was able to gather that the proof involves a generalization of Euler's famous formula \(e^{i\pi}=-1\), which is partly only a thing due to using π instead of τ. (The τ version, \(e^{i\tau}=1\) is equally true but incredibly basic—it essentially says that if you go around \(360^\circ\) then you've made a full circle—and hasn't caught people's imagination the way Euler's version has.)

I've really only been able to do a bare minimum of reading about this topic, but I'll definitely be keeping an eye on in the coming days and try to update you as I learn more. (The possibility—however remote—of one's Ph.D. being for naught is incredibly motivating!) A hui hou!