Science & Space · Mathematics

The Riemann Hypothesis

The most famous open problem about prime numbers, explained simply, plus the October 2026 AI proof of the quasi-Riemann hypothesis

Prime numbers are the whole numbers that can't be split into smaller whole-number factors: 2, 3, 5, 7, 11, 13, and so on forever. Every other whole number is built by multiplying primes together, which is why mathematicians call them the atoms of arithmetic. The primes thin out as you count higher, but they never stop, and they never fall into a simple pattern. The Riemann hypothesis is the best guess anyone has ever made about exactly how messy that pattern is allowed to be.

The Problem in Plain Language

In 1859 Bernhard Riemann found a function, now called the Riemann zeta function and written ζ(s), that secretly encodes the primes. You feed it a number and it gives a number back. The inputs it cares about are complex numbers, which you can picture as points on a flat map with an east-west coordinate (the "real part") and a north-south coordinate (the "imaginary part").

For real inputs bigger than 1, the formula is just an infinite sum: ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + … A clever extension lets you plug in almost any point on the map. The interesting question is where on the map ζ(s) equals zero. Some zeros are easy and boring (at −2, −4, −6, and so on). All the others, called the nontrivial zeros, sit inside a vertical band called the critical strip, where the east-west coordinate is between 0 and 1.

The Riemann hypothesis says every nontrivial zero sits exactly on the center line of that strip, where the real part is 1/2. Computers have checked trillions of zeros and every one is on the line. Nobody has a proof that all infinitely many are.

Kontorovich's "signal processing" picture

On the podcast, Kontorovich (Rutgers) explained it as signal processing on the primes (transcript, around 1:06:36). A microphone turns your voice into a list of frequencies and amplitudes, and a speaker rebuilds the voice from that list. The zeta function does the same thing for the primes: its zeros are the frequencies, and adding those waves back together reconstructs where the primes fall. In his words, the Riemann hypothesis predicts that "the orchestra that plays the music of the prime numbers has every single instrument sounding at exactly the same amplitude" (1:08:57).

Why It Matters

• It controls how the primes are spread out. The prime number theorem (1896) tells you roughly how many primes there are up to any size. The Riemann hypothesis would guarantee the error in that estimate is as small as it could possibly be.
• Thousands of results depend on it. Number theorists routinely prove theorems that start with "if the Riemann hypothesis is true." Kontorovich said on the episode that it "has a thousand applications in very faraway fields" (around 1:20:01).
• It is a Millennium Prize Problem. The Clay Mathematics Institute offers $1 million for a proof. It was also on David Hilbert's famous 1900 list of problems.
• It is the easiest hard question to state. Steven Brunton put it this way on the episode: adding and multiplying are the two simplest ideas in math, and "we don't know how they work together" (around 1:21:46). Primes are where that mystery lives.

The October 2026 Breakthrough: Quasi-RH

The full hypothesis says the zeros are all on the line at 1/2. A quasi-Riemann hypothesis is a weaker statement: it says the zeros are at least kept away from the edges of the strip, inside some narrower band. Before October 2026, mathematicians only knew a zero-free region that got thinner and thinner the higher up the strip you went. According to Kontorovich on the episode, "that thickness hasn't moved in seventy years" (1:26:00).

The AI proof does something much stronger. According to Kontorovich, it shows there are "no zeros with real part between seven-eighths and one, an entire infinite vertical strip" (1:26:00). By symmetry, that also clears the strip between 0 and 1/8. The proof covers not just the zeta function but all Dirichlet L-functions, its close relatives (1:10:27).

What the guests said

• Shock. Kontorovich: "I did not think I would see a proof of the quasi-Riemann hypothesis in my lifetime. So the fact that I saw it on Tuesday was extraordinarily shocking" (01:09:45). He was at the National Museum of Mathematics gala in New York when the results dropped.
• Confidence from Lean. The statement was already written in Mathlib, the shared library of computer-checked math, and the proof checks in Lean with no gaps ("sorry-free"). Kontorovich said that gives him "100% confidence that the proof is right" (01:23:59).
• Old tools, new combination. At a high level, Kontorovich described the ideas as standard analytic number theory, such as embedding the Möbius function and twisting it by sextic characters, "chained in just the right way" (01:23:59).
• Not the full thing. There is a second proof reaching about 11/12, and Kontorovich said that method "sort of fundamentally can't get down to one half" (01:26:34). Getting from 7/8 to 1/2 would be the full Riemann hypothesis, and he said he doesn't know whether humans or AI will get there.

Outside the episode

Quanta Magazine reports that OpenAI released two versions: a 199-page, Lean-verified proof of the 7/8 bound dated September 30, 2026, and a shorter 49-page proof of the 11/12 bound that most experts are reading first. Quanta also reports that the experts who have studied it largely agree it is correct, and that its novelty is in how it combines existing ideas. The papers are in OpenAI's math repository.

Resources

• Lex Fridman Podcast #503: AI Crisis in Math (Moitra, Brunton, Kontorovich)
• Episode #503 transcript (Riemann hypothesis chapter starts 1:05:56)
• Episode #503 on YouTube
• Clay Mathematics Institute: The Riemann Hypothesis
• Wikipedia: Riemann hypothesis
• Quanta Magazine: AI Proved the Quasi-Riemann Hypothesis. What Happens Now? (Oct 19, 2026)
• OpenAI: Sharing AI progress in mathematics (Oct 6, 2026)
• OpenAI preprint: The Quasi-Riemann Hypothesis (11/12 version, PDF)