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 ζ(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 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 wrote in the paper laying out the Millennium Prize formulation of the problem: “The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers.”
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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.
OpenAI announced two different versions of their quasi-Riemann hypothesis proof. The first version, dated September 30, 2026, 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 (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.
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Mathematicians have been studying the paper intensely in the couple of weeks since its release. Peter Sarnak, a prominent number theorist at Princeton University, has decided to repurpose a course he’s teaching this semester to study it. Radziwiłł says he can now sketch the proof’s arguments in half a page, while more involved write-ups by other mathematicians range from seven to 13 pages. (As of this writing, these have not yet been published, but are circulating privately.) These experts largely agree that the result is correct — and that its novelty lies in the way it brings together ideas from number theory that have been around for some time.
I never cared for math but this sent me down a rabbit hole. I've got some questions that are half legitimately asking and half making sense of things in my head. So the Riemann Hypothesis is...
I never cared for math but this sent me down a rabbit hole. I've got some questions that are half legitimately asking and half making sense of things in my head.
So the Riemann Hypothesis is important because it reflects the fundamental nature of primes, and proving it would prove 1000s of other math theorems?
Also, 1/2 is needed to zero out the zeta function with ALL primes?
If we understand this fundamental nature of prime numbers, would this let us guess or calculate all primes? Or we would just know what prime numbers cannot be?
Or are prime numbers by definition impossible to completely predict? I saw that Euclid proved that there's an infinite number of primes because you can just multiply two primes together and add 1 (2*5+1=11)
Once again, I do not normally care about math but somehow this Riemann hypothesis got very interesting to me lol
Well, math is deterministic. There is a sense in which we could, in principle, calculate whether any number is prime. We already do this for numbers that aren’t extremely large. But being able to...
Well, math is deterministic. There is a sense in which we could, in principle, calculate whether any number is prime. We already do this for numbers that aren’t extremely large.
But being able to calculate something in principle doesn’t mean we can do it in practice for all numbers. There will always be numbers that are too big to manipulate using any computer we could possibly build.
Theoretical math is about what we can say about numbers that are too large to calculate. Testing a few trillion numbers tells us something, but you never know if there will be a counterexample that’s just beyond the numbers we tested. This is why it’s done by proving theorems rather than doing calculations.
Primes are really important. It sounds like a curiosity when you learn it in primary school. But when you take abstract algebra in college, then you realize just how fundamental primes are. The...
So the Riemann Hypothesis is important because it reflects the fundamental nature of primes, and proving it would prove 1000s of other math theorems?
Primes are really important. It sounds like a curiosity when you learn it in primary school. But when you take abstract algebra in college, then you realize just how fundamental primes are. The element comparison is apt - what are elements? The indivisible components of matter. All things are composed of some combination of elements. Similarly, all integers are composed of primes. That can be seen from a relatively simple proof by contradiction.
It also tells us a lot about the Reimann Zeta function, which itself is very important in real and complex anal.
Also, 1/2 is needed to zero out the zeta function with ALL primes?
The Zeta function does not directly relate to primes, it is conjoined by the Euler product.
If we understand this fundamental nature of prime numbers, would this let us guess or calculate all primes? Or we would just know what prime numbers cannot be?
In essence, yes, although it's not particularly that property that is important. Even a middle schooler can tell what is and isn't a prime by hand.
From the article:
[...]
[...]
I never cared for math but this sent me down a rabbit hole. I've got some questions that are half legitimately asking and half making sense of things in my head.
So the Riemann Hypothesis is important because it reflects the fundamental nature of primes, and proving it would prove 1000s of other math theorems?
Also, 1/2 is needed to zero out the zeta function with ALL primes?
If we understand this fundamental nature of prime numbers, would this let us guess or calculate all primes? Or we would just know what prime numbers cannot be?
Or are prime numbers by definition impossible to completely predict? I saw that Euclid proved that there's an infinite number of primes because you can just multiply two primes together and add 1 (2*5+1=11)
Once again, I do not normally care about math but somehow this Riemann hypothesis got very interesting to me lol
Well, math is deterministic. There is a sense in which we could, in principle, calculate whether any number is prime. We already do this for numbers that aren’t extremely large.
But being able to calculate something in principle doesn’t mean we can do it in practice for all numbers. There will always be numbers that are too big to manipulate using any computer we could possibly build.
Theoretical math is about what we can say about numbers that are too large to calculate. Testing a few trillion numbers tells us something, but you never know if there will be a counterexample that’s just beyond the numbers we tested. This is why it’s done by proving theorems rather than doing calculations.
Primes are really important. It sounds like a curiosity when you learn it in primary school. But when you take abstract algebra in college, then you realize just how fundamental primes are. The element comparison is apt - what are elements? The indivisible components of matter. All things are composed of some combination of elements. Similarly, all integers are composed of primes. That can be seen from a relatively simple proof by contradiction.
It also tells us a lot about the Reimann Zeta function, which itself is very important in real and complex anal.
The Zeta function does not directly relate to primes, it is conjoined by the Euler product.
In essence, yes, although it's not particularly that property that is important. Even a middle schooler can tell what is and isn't a prime by hand.