qedbot

The Millennium Museum·Analytic number theory

Open

The Riemann Hypothesis

Do all the non-trivial zeros of the Riemann zeta function lie on the critical line, where the real part is one half?

Open. In August 2026 an AI system raised the proven share of zeros on the critical line to 67.2%, which is progress rather than a proof.

The exhibit

The problem, three ways

Curious

The primes look random, but how they are spread out is governed by the zeros of a single function, the Riemann zeta function. In 1859 Bernhard Riemann guessed that those zeros all sit on one straight line. If he was right, the primes are as regular as they could possibly be. Nobody has proved it.

Undergraduate

The series ζ(s) = Σ n⁻ˢ converges for Re s > 1 and extends to the whole complex plane except s = 1. Its zeros at the negative even integers are called trivial; all the others lie in the strip 0 < Re s < 1. The hypothesis says they all have real part exactly 1/2. It is equivalent to the error in the prime number theorem being as small as O(x^(1/2+ε)).

Specialist

A positive proportion of zeros lie on the line (Selberg), raised by mollifier methods from Levinson's third to Conrey's two fifths and Pratt, Robles, Zaharescu and Zeindler's five twelfths, and in 2026 to 67.2% by an argument found by an AI system. The generalised forms for Dirichlet and automorphic L-functions are separate statements; a proof for ζ need not settle them.

What would count

  • A proof that every non-trivial zero of ζ has real part 1/2, or a single rigorously verified zero off the line.
  • Clay's conditions: publication in a qualifying outlet, two years, general acceptance, then a decision by its board.

Fidelity traps

  • A result about a proportion of the zeros is progress, not the hypothesis.
  • The generalised Riemann hypothesis for other L-functions is a different problem.
  • A formal proof should prove the stated hypothesis itself, not a variant carrying extra assumptions.

Who is attacking it

  • Anthropic: A Claude model raised the proven proportion of zeros on the critical line to 67.2%, with a Lean formalisation rebuilt by qed.bot. Source
  • OpenAI: Says it began large-scale work on all the open Millennium problems on 1 September 2026. Source

The history

11 events
  1. 19th century
  2. 1859-11

    posed

    Riemann's paper on the primes

    In an eight-page paper on the number of primes below a given size, Riemann extends the zeta function to the complex plane and remarks that it is very probable all its non-trivial zeros have real part one half.

    Source

  3. 1896

    progress

    The prime number theorem

    Hadamard and de la Vallée Poussin independently prove the prime number theorem by showing that ζ has no zeros on the line Re s = 1.

    Source

  4. 1900s
  5. 1900

    posed

    Hilbert's eighth problem

    Hilbert includes the Riemann hypothesis in his list of problems for the new century.

    Source

  6. 1910s
  7. 1914

    progress

    Infinitely many zeros on the line

    Hardy proves that infinitely many zeros lie on the critical line.

    Source

  8. 1940s
  9. 1942

    progress

    A positive proportion

    Selberg proves that a positive proportion of the zeros lie on the critical line.

    Source

  10. 1970s
  11. 1974

    progress

    More than a third

    Levinson shows that more than one third of the zeros lie on the line.

    Source

  12. 1980s
  13. 1989

    progress

    More than two fifths

    Conrey raises the proportion on the line to more than two fifths.

    Source

  14. 2000s
  15. 2000-05-24

    prize

    A Millennium Prize Problem

    The Clay Mathematics Institute names the Riemann hypothesis one of seven problems carrying a $1,000,000 prize, with an official description by Enrico Bombieri.

    Source

  16. 2004

    computation

    Ten trillion zeros checked

    Gourdon verifies by computer that the first 10¹³ zeros lie on the critical line.

    Source

  17. 2020
  18. 2020

    progress

    More than five twelfths

    Pratt, Robles, Zaharescu and Zeindler show that more than five twelfths of the zeros lie on the line.

    Source

  19. 2026
  20. 2026-08-10

    AI

    An AI system reaches 67.2%

    A Claude model, asked to attempt the hypothesis itself, raises the proven proportion of zeros that are simple and on the line from 41.6% to 67.2%. The proof is formalised in Lean, and qed.bot rebuilt it: 23 theorems on the standard axioms.

    Source · In the register

  21. Next
  22. ?

    The next entry

    Follow this problem below to get an email the moment it moves.

Follow and discuss

All discussion

Discussion and bounties for this problem load here.