AI has purported to prove two new and improved versions of the quasi-Riemann hypothesis, as discussed at Quanta magazine and a Twitter thread discussing additional proofs (by humans) that flow from the new quasi-Riemann hypothesis proof.
What is the Riemann Hypothesis?
The Riemann hypothesis can be explained as follows (from the first link):
The Riemann hypothesis is about which values of s will make the zeta function equal zero.
But it is also about far more than how this particular function behaves. As Bernhard Riemann realized when he first made the conjecture in 1859, the function relates to patterns in the prime numbers — integers divisible only by 1 and themselves that many mathematicians call the “atoms of arithmetic.”
It’s long been known that as you move along the number line, the primes thin out. Riemann showed that exactly how the primes thin out is, in a sense, governed by the zeta function — in particular by the places where it equals zero.
It’s straightforward to prove that if s is a negative even number (−2, −4, −6, …), then ΞΆ(s) = 0. These are called trivial zeros. The Riemann hypothesis says that the only other zeros of the zeta function, called nontrivial zeros, occur when s has a real part of 1/2. That is, they should all be located on a vertical line:
If any nontrivial zero were to be found somewhere away from that vertical line, it would imply that the primes must form an extremely surprising pattern, one that has never been observed to date.
Just narrowing the area where the nontrivial zeros can appear has had huge ramifications. Riemann himself proved that nontrivial zeros of the zeta function can only be found in a vertical strip between zero and 1. Then, in 1896, two mathematicians showed that zeros could not lie on the edges of the strip. As a direct consequence, they were able to prove an important result called the prime number theorem, a formula for how quickly, on average, the primes thin out.
If the Riemann hypothesis holds, then mathematicians can guarantee that any deviations from the average given by the prime number theorem will always be small.
Mathematicians have numerically verified(opens a new tab) that the first 12 trillion nontrivial zeros are on the 1/2 line. But they can’t be sure the hypothesis holds for all the zeros, and even one solitary outlier would land like an earthquake. As Enrico Bombieri(opens a new tab) wrote in the paper(opens a new tab) laying out the Millennium Prize formulation of the problem: “The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers.”
The connection between the zeta function and the primes was known before Riemann. In 1737, Leonhard Euler showed that for any complex number s, the zeta function is equivalent to another function that can be written entirely in terms of primes:
This brings us to the quasi-Riemann hypothesis. Where the Riemann hypothesis posits that all the nontrivial zeros are on the vertical line at 1/2, the “quasi” version just asks one to prove that they are all contained in a vertical strip that’s narrower than the known strip between zero and 1. The narrower the strip, the more one can say about how the primes are distributed.
What did AI recently prove?
OpenAI announced two different versions of their quasi-Riemann hypothesis proof. The first version, dated September 30, 2026(opens a new tab), claimed to show that all nontrivial zeros must be between 1/8 and 7/8, and was formally verified using the Lean programming language. The second version, which was completed almost a week later(opens a new tab) (and is not yet formally verified), claimed to prove a weaker result: that the zeros must have a real part between 1/12 and 11/12. But while the first paper is 199 pages long, the second weighs in at a comparatively modest 49 pages.
More background
The Euler version makes it more obvious, but the zeta function is undefined and at s=0, and approaches a limit of infinity close to s=0.
Even this simple explanation for a relatively simple hypothesis (all the good ones are) begs the question of what a complex exponent of an integer even means (and I won't try to explain that here). Suffice it is say that this isn't trivially intuitive in the way that an integer exponent is, and that it is less intuitive than a non-integer real exponent is.
Commentary on the AI proof
First off, consider that the new proofs are less impressive than they seem. We already knew that the real part of the exponent s in a zeta function zero has to be more than 0 and less than 1. We knew that in 1896, one hundred and thirty years ago. Narrowing that range to greater than 1/12 and less than 11/12, or greater than 1/8 and less than 7/8 improves on that result, but it isn't that Earth shaking of an improvement over the status quo.
Ideas for breaking down the Riemann Hypothesis into subparts
Next, lets break this down and consider some points that would help prove the overall Riemann hypothesis. Some of them may already have been proven. All of them must be true if the Riemann Hypothesis is correct, which numerical methods strongly suggest.
* It would be nice to prove that for any given value of the imaginary part of s that there is only one zero.
* It would be nice to prove that for any given value of the real part of s that there are either zero or an infinite number of zeros.
* It would be nice to prove that, to produce a zero, the real part of s must be a rational number (a subset of the real numbers).
* It would be nice to prove that, to produce a zero, the imaginary part of s must be a whole number (i.e. a positive, non-zero, integer), or failing that, that it must be a positive rational number, or in between, a positive rational number related by some rule to the rational number that is the real part of s, or better still, a whole number related by some rule to the rational number that is the real part of s.
* It would be nice to prove that the imaginary part of s must be an odd number to produce a zero.
* It would be nice to prove that all values of s with a real part of 1/2 and an imaginary part which is a prime number of zeros. Keep in mind that the Riemann Hypothesis is an if and only if conjecture: it says that non-trivial zeros have a real part of 1/2 and an imaginary part that is prime, and it also says that there are no other non-trivial zeros. Proving the affirmative part standing alone would be a big deal, even without a proof that there are no other non-trivial zeros, but requires consideration of far fewer possibilities.
* Proof of the affirmative part of the Riemann Hypothesis would likely require that the term in the Euler infinite series definition of the zeta function where integer in the imaginary part of s is equal to the number that it is the exponent of is special somehow, because it cancels out all of the other terms of the infinite series (I don't think that a single term of either infinite series definitions of the zeta function can be zero by itself, but proving that would itself be a step).
* Proof of the negative part of the Riemann Hypothesis might start with a determination that the imaginary part of s can't produce a zero if it is even.
* Proof of the negative part of the Riemann Hypothesis might establish that the imaginary part of s can't produce a zero if it has exactly two, or better yet, an even number, of prime factors.
* Proof the the negative part of the Riemann Hypothesis might establish that the imaginary part of s can't produce a zero if it has an odd number of prime factors greater than one.
* Proof of inability to produce a zero, in a given case, amounts to showing there are one or more terms of the infinite series that don't cancel out at that value.
* It would be nice to tie in the conclusion about trivial zeros of the zeta function to the conclusion about non-trivial zeros of the zeta function somehow.
Post Script
Of course, I'm not going to prove (or disprove) the Riemann Hypothesis. Complex analysis (i.e. the calculus of imaginary numbers) was my weakest subject as a math major in college, and I never formally studied number theory at all.
And, I only scratch the surface of the most obvious ideas in a deeply reached and explored corner of mathematics.
But I had a long dentist appointment this morning and needed a distraction, so I thought about it then, and that's why I'm writing about it now.
