Short answer: you can’t. But the setup gets close enough that it takes real work to find the exact reason it can’t — and that reason turns out to be more interesting than the failed attempt itself.
The setup
The Riemann Hypothesis says every non-trivial zero of the zeta function sits on a specific vertical line in the complex plane. Nobody has proven it, but there’s a beautiful conjectural shortcut called the Hilbert–Pólya program: if you could find some self-adjoint operator — the kind of thing that shows up constantly in quantum mechanics, always with real eigenvalues — whose eigenvalues happened to be exactly the zeta zeros, the Hypothesis would fall out for free. Self-adjoint operators can’t have complex eigenvalues, so if the zeros are a spectrum, they’re automatically on the line.
The catch, for over a century, has been finding the operator. Nobody has one. But the conjecture is specific enough that you can go looking, and people have. This project is one of those searches, built on an unusually elegant piece of geometry: the Hilbert curve.
A Hilbert curve is a way of snaking a single path through every cell of a grid — 2D, 3D, or higher — so that consecutive integers always land on touching cells. It’s a real, well-studied mathematical object, mostly famous for spatial indexing (databases use it to keep nearby data physically nearby on disk). The idea here: color the cells by whether their position along the curve is a prime number, slice the grid into layers, and measure how much each layer’s prime density correlates with its neighbors. That gives you a matrix. Matrices have eigenvalues. Do the eigenvalues look like zeta zeros?
They do. Suspiciously well.
The suspiciously good part
Take the first 64 known zeta zeros — real numbers, computed to high precision, publicly available. Take the eigenvalues of this Hilbert-curve matrix, sorted by size. Line them up and compute a correlation coefficient. You get numbers like −0.99. In statistics, that’s about as strong as correlation gets. A properly randomized permutation test (shuffle the eigenvalues at random 20,000 times, see how often you’d get a match this good by chance) says: basically never. This isn’t noise.
And there’s a genuine, provable piece of mathematics sitting underneath it. If you build the layer-comparison matrix using true spatial adjacency — literally, which grid cells physically touch — instead of just following the curve’s path, something clean happens: the matrix becomes exactly tridiagonal. Every entry more than one step off the main diagonal is precisely zero, not approximately, to floating-point precision, for every dimension and every size tested. That’s a real theorem with a one-paragraph proof: a face-touching neighbor changes exactly one coordinate by exactly one step, so it can only ever land you in an adjacent layer or the same one, never two layers away. Nothing about primes is needed for that part — it’s just geometry.
And tridiagonal matrices with constant diagonals have been fully solved since the 1950s. Their eigenvalues are a simple cosine formula. You don’t even need a computer to diagonalize them — you can write down the answer. So this Hilbert curve operator isn’t just numerically close to the zeta zeros; it has an exact, closed-form spectrum, and that spectrum is what’s correlating at −0.99.
Where it falls apart
Here’s the thing about a cosine. It oscillates between a fixed high point and a fixed low point, forever. No matter how big you make the matrix — double it, square it, go to a million dimensions — the eigenvalues stay trapped in the same bounded range. You just get more of them, packed more tightly into the same box.
The zeta zeros do not do this. They keep growing, forever, and the rate they grow at is known precisely — it’s roughly (T/2π)log(T/2π), a formula proven a century ago. It’s not just that the zeros get bigger; the spacing between them shrinks in a very specific, logarithmic way as you go further out.
A bounded, oscillating spectrum and an unbounded, log-growing spectrum are different kinds of thing. You cannot rescale one into the other with a multiplier, a polynomial, or any calibration curve, because rescaling preserves boundedness. If your sequence lives in a box, no formula turns it into a sequence that runs off to infinity. This is the actual reason the correlation, however strong, can never become an equality. Not “we haven’t found the right scaling yet” — a proof that no scaling exists.
The −0.99 correlation, it turns out, is really a curve-fitting accident: a cosine and a k log k-shaped curve happen to look a lot alike over a short enough stretch — which the first 64 zeros are. Taylor-expand the cosine near its peak and you get something quadratic; the zeta counting function, over that same short range, is also well-approximated by something close to quadratic. Two different functions, briefly wearing the same disguise.
Not the first time
This is, gratifyingly, not a unique or embarrassing failure — it’s a recognizable type of failure, and other serious attempts hit variations of it:
- Wu–Sprung (1993): physicists built a one-dimensional quantum potential, via genuine inverse-scattering mathematics, whose energy levels reproduce the first five hundred zeta zeros almost exactly. The catch: they admit outright that you can fit any finite stretch of zeros this way by tuning the potential further — it’s a fit to a known answer, not a derivation from one.
- Berry–Keating (1999): the H=xp model gets the growth rate right, by a genuine physics argument about phase space volume — but the operator itself isn’t rigorously well-defined without extra assumptions nobody’s proven.
- Connes: the noncommutative-geometry approach sidesteps the whole density problem with much heavier machinery, and lands the Riemann Hypothesis exactly where it reduces to another open conjecture.
Every serious attempt seems to get exactly one or two of the three things it needs — correct density, rigorous operator, actual proof — and never all three at once.
Why bother, then
Because the process of finding out exactly why something plausible fails is not the same as it just failing. You come out the other side with a real theorem — the tridiagonal structure, useful independently as a cheap way to verify a space-filling curve implementation isn’t secretly broken, which, during this project, it was, twice — and a precise criterion anyone else can check their own candidate operator against before sinking months into it.
The Riemann Hypothesis is still open. This didn’t move the needle on it. But it drew one clean, small circle around a place a proof cannot live, and that’s a legitimate thing for a piece of math to do, even when the headline is “no.”