I’m not a mathematician, but GPT is helping me to become one.
Let’s start with an assertion that I saw somewhere on the web, which is that a certain company is trying to “solve” something called the Riemann hypothesis, indicated in the Millennium Prize question set (I’ll get to that later), using AI in some capacity.
I saw the name, ‘Riemann hypothesis’ and my curiosity was piqued. It sounded important, like Moore’s Law, or Godwin’s Law, but the thing is, I don’t know a flying fig about any of this, and Wikipedia wasn’t much help. Here’s the wiki definition paragraph:
“In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.”
Other than that last sentence telling me what egghead thought this up, it’s all Greek to me. So I did what I have become accustomed to doing: I asked GPT, then asked it to simplify, and got this:
“The Riemann Hypothesis is a famous claim about a special math function. It says all the important places where that function equals zero line up neatly on one vertical line. If it’s true, we’d understand the pattern of prime numbers much better.”
That’s much better. Here’s some more.
Q: how would we understand primes better? why?
“Because primes are ‘encoded’ in the zeta function. Mathematicians have formulas that connect how many primes are ≤ N to the zeros of the zeta function. Those zeros create the ‘wiggles’ (error) between our smooth estimate of prime counts and the real count. If the Riemann Hypothesis is true, it says those zeros are in the nicest possible position, which forces the wiggles to stay small. So we’d get much tighter, reliable bounds on how irregular prime spacing can be—i.e., prime-counting estimates become far more precise, and lots of ‘if RH then…’ theorems in number theory become unconditional.”
From this I’m gleaning the concept of nested hypotheses, which just seems insane to the lay person, but I’m still confused about what you need to do to “solve” this. And before I go any further, here’s what the stakes are.
Playing to Win
It turns out that a math institute is offering anyone a million dollars to “solve” the Riemann hypothesis.
An actual million dollars.
Do I have your attention now?
As part of the Millennium Prize problem set, the Riemann Hypothesis has been selected by the Clay Institute which has set up this generous payout to anyone who can do what hasn’t yet been done. Some have tried: some guy presented something in 2018 at a conference. But it didn’t take. It wasn’t good enough.
Speaking of “good enough,” let’s get back to the issue at hand. Why can’t people solve the Riemann Hypothesis? What do we lack? And how could AI help?
More Questions
I asked GPT:
Q: what does it mean to “solve” it? Does it mean prove or disprove?
A: “Yes. ‘Solving’ the Riemann Hypothesis means settling the statement decisively:
Prove it: Give a rigorous mathematical proof that every nontrivial zero of the zeta function has real part ½. Disprove it: Find one nontrivial zero that’s not on that line (a counterexample), and prove it really is a zero. Either one would count as a solution—proof or disproof.”
Q: why is that not possible?
A: “It’s not that it’s “not possible” — it’s just not solved yet because the tools we have haven’t been strong enough.”
That sentence right there implies that AI will help us to build the stronger analysis that we need to crack the case on the Riemann Hypothesis. But what are the challenges? GPT also provided this helpful bullet point list for “why it’s hard.”
· It’s an infinite claim. You can’t check “enough” zeros; you need an argument that covers all of them.
· Zeros are slippery. The zeta function lives in the complex plane and is tied to oscillating sums/products; small analytical gaps can ruin a proof.
· Partial progress isn’t enough. We can prove many zeros lie on the 1/2 line and that all zeros lie in a certain strip, but pushing that to 100% has resisted every known method.
· It sits at a crossroads. Any proof likely needs a new idea connecting complex analysis, harmonic analysis, and number theory more tightly than current techniques.
Zeroes are slippery. Ok. That sounds like something a mathematician would say.
Along with this: “So the challenge isn’t ‘can we find lots of zeros?’—we can. It’s “can we cover all zeros, forever, with absolute certainty?” That’s a proof problem, not a computation problem.”
Q: if it’s infinite, how would AI help?
A: “AI can’t “check infinitely many zeros” by brute force. If AI helps, it would be by helping humans find a proof idea that covers all zeros at once.”
Q: how would there be a proof that covers all zeroes if the zeroes are infinite?
A: “A proof doesn’t list the zeros. It proves a rule that forces every zero, no matter how high, to behave. Think of it like this: there are infinitely many even numbers, but you can prove ‘every even number is divisible by 2’ with one argument. So the proof would show: ‘If a zero existed off the 1/2 line, it would contradict property P.’ Since property P holds everywhere, no off-line zero can exist—covering infinitely many in one shot.”
I found this to be pretty capable, really, the idea of tying it to something like “are all even numbers divisible by two” but there’s still something here that is showing how mathematics becomes philosophical at its core. We will never plumb infinity, so how can we solve infinite puzzles? Isn’t a hypothesis, at the end of the day, just a hypothesis? How do we know, in mathematics or elsewhere, is we will ever find an exception? I’m going to show you more of this that covers that question and what it means to “solve” something like this.
Q: but we can’t check for that? we can’t see whether there is an exception or not?
A: “Right. You can’t reliably “look for an exception” the way you can for a finite list. A disproof is easy in principle: find one zero off the 1/2 line and verify it rigorously. A proof can’t come from checking cases, because no matter how far you check, there are always more zeros above that. We can check huge ranges computationally… but that still leaves infinitely many heights untested, and an exception, if it exists, could be astronomically far up.”
Me: this sounds like absolute nonsense.
A: “I get why it feels that way. The key is: in math, ‘knowing’ isn’t the same as ‘checking.’ You can check a lot of cases and build confidence. But that never becomes certainty for an infinite set. Example: I can check a million numbers and see they’re not equal to 0. That still doesn’t prove ‘no number is 0,’ because I didn’t check all numbers.”
Q: ok, here’s my question about the Riemann hypothesis then: if we have a colossal number of checked cases up the number line, and we know the set is infinite, and we haven’t found an exception, can’t we “call it good” and suggest that there’s not an exception somewhere else? Is there no theorem that covers additional cases up the line (beyond our knowledge set)? Isn’t that how science usually works?
Really pay attention to GPT’s answer here.
“You can ‘call it good’ in an engineering / scientific sense. Lots of people do. But math is playing a different game … Science is comfortable with induction (‘this has held in every experiment, so we accept it provisionally’). A mathematical theorem is deductive: it must be impossible for an exception to exist anywhere.”
One More Thing: The Role of Artificial Intelligence
I hope that the above shows, to an extent, what humans are wrangling with here.
I wanted to include just one more of GPT’s longer answers to the question: what would AI do for this?
“AI helps by boosting the parts that aren’t ‘check infinitely many cases’.”
Part of that, GPT said, is finding the right intermediate target, and “suggest(ing) new lemmas/inequalities,” where the model notes:
“Many RH approaches boil down to proving a very sharp bound or positivity property. AI can comb through related proofs and propose candidate inequalities or missing steps humans might not try.”
Then there’s also the promise of proof assistance and bug finding, as well as literature synthesis, where work on this subject spans many subfields.
If you’ve got this far with me, your takeaway may be this: ‘I’m not a mathematician.’ But GPT breaks it down to where we can at least see what the mathematician is doing. We can see the quandary, and why this problem remains unsolved, and, to an extent, what AI may be able to do about it.
That million dollars is still out there. Get to it.
