If you keep an eye on our updates, you may know already that yesterday was the day when Riemann hypothesis (RH) was going to be proved by British-Lebanese Mathematician Sir Michael Atiyah at an event. The event was Heidelberg Laureate Forum (HLF) 2018, and Sir Atiyah provided his proof during a 45-minutes long talk, as promised. We had written a detailed article on what RH is what impact its proof may have on cryptography. Now when it has finally been proved, we’re going to tell you about how it has been proved and whether it can really impact cryptography or not. Let’s get started.
A simple proof
Sir Atiyah said that he has found a “simple self-contained proof” for Riemann hypothesis. So from the very beginning, things were looking very suspicious. There can’t be a simple self-contained proof for something that remained unproved for as long as 150 yrs. Had there been any such proof, it would’ve been found till now. And to be honest, had anyone else claimed to have found a self-contained proof for this hypothesis he would’ve been considered an idiot by the majority of people. But since it’s Sir Michael Atiyah – the winner of Fields Medal (1966) and Abel Prize (2004) – he couldn’t be taken lightly. However, there are speculations that he has become less careful with age (he’s 89 yrs old), so we can’t be too sure about his own work even. People do mistakes.
Proof by accident
As if things were not skeptical enough already, Sir Michael Atiyah made them more skeptical by declaring that he found the proof accidentally!
That’s true. According to him he had not set out to prove RH. Instead, he was trying to derive the value of fine structure constant in Physics. RH was only a bonus!
What a pleasant accident, isn’t it?
The Todd function
We don’t know much about this function as of now, except for the fact that it’s the key to Riemann hypothesis proof provided by Sir Atiyah. According to him it’s “a very very clever function that maps Euler’s equation to its quaternionic generalization, and is defined by an infinite iteration of exponentials”. The function is named after his teacher J.A. Todd, and it has not yet been published. It’s a weakly analytic function of a complex variable s≠0. For real values of s > 0 the value of this function increases from T(1) = 1 to T(π) = ж.
This means that on the critical line of Riemann zeta function the Todd function has a limit of ж, and the fine structure constant α = 1/ж. That is:
limy → ∞ T(1/2 + yi) = ж = 1/α = 137.035999…..
Now, there’s a caveat with all this stuff. The fine structure constant is measured empirically – not in exact mathematical terms. So we don’t know why he was trying to prove that a physical constant has an exact mathematical value.
Proof by contradiction
Now let’s come to the meat of the matter. In logic, proof by contradiction is a kind of indirect proof that begins by assuming that opposite of something is true. If that assumption leads to a contradiction, the original proposition is considered proved. And this is the kind of proof that Sir Atiyah used to prove Riemann hypothesis. Have a look on the slide below:
You see that? That’s the simple self-contained proof found by Sir Michael Atiyah for RH.
So, what about the Millennium prize?
When asked by the audience whether Sir Atiyah would claim the million dollar prize on Riemann hypothesis by Clay Mathematics Institute, he answered “YES”. He said that he’ll claim the prize because he deserves it.
Could Atiyah’s proof be wrong?
If he hasn’t shown it to anyone, then most probably it’s wrong. That is because in mathematics no one does flawless work by himself. So there’s a chance that his work may have a flaw. Now, if that happens then it will be interesting to see whether the flaw is patchable or unpatchable. If patchable then someone will fix it and Atiyah’s proof of Riemann hypothesis may remain valid. If not patchable then Atiyah will be discredited.
The Clay Mathematics Institute (CMI), which will reward a million dollars to Atiyah for proving this hypothesis, has also not taken the proof as valid for now. In fact, the bar set by them to consider something valid is a lot higher than that of any other mathematician – not only the proof is required to be reviewed by peers, it should also be published in a reputed journal for more than 2 yrs before it can be considered valid. This means that we won’t come to know for at least 2 years whether Sir Atiyah is right or wrong.
Could this proof be used to break modern encryption?
We don’t know yet. And that is because we still don’t know whether his proof is valid or not. Unless his proof is examined by the peers it can’t be considered valid. And if some flaws are found in it then the validation is going to take a looonnggg time. That is because in mathematics and computing the duration of fixing any patchable flaws can be too long. An example is Fermat’s last theorem, which had a flaw that could be patched after hard work of a whole year.
In short, everything is very uncertain as of now. We can’t say whether Sir Atiyah’s proof is valid or not, and therefore we can also not say whether his proof can be used to break modern cryptography or not. Everything will be clear in future, so we should probably wait and watch what happens next.