Riemann Hypothesis
Do all non-trivial zeros of the zeta function lie on one line? Open since 1859; it governs how primes are distributed.
Open in the interactive tree →Bernhard Riemann conjectured in 1859 that all non-trivial zeros of the zeta function have real part one half. This would determine how evenly the primes are spread, and hundreds of theorems are currently proven only 'assuming' it is true. More than 10^13 zeros have been checked numerically without exception, which is not a proof. Clay prize: one million US dollars.
As of October 2026
October 2026: unproven. Progress comes in partial results: in 2024 Guth and Maynard improved bounds on how many exceptional zeros could exist. On 10 August 2026 Anthropic reported that an internal research version of Claude proved that more than 67.2% of all non-trivial zeros are simple and on the critical line (previous record about 41.6%), checked by Anthropic mathematicians and outside experts and formalized in Lean; Lamzouri posted a shorter proof on 2 September 2026 (arXiv 2609.02882). That is still far from 100%.
What is missing
- An idea that captures all zeros at once: current methods (Levinson, mollifiers) give only proportions, never 100%
- A convincing spectral explanation (Hilbert-Pólya idea, random matrices): a suitable operator has not been found
- Computation cannot prove it: a counterexample is possible in principle but has never appeared
- A synthesis of analysis, number theory and physics (random-matrix statistics)
Becomes possible once solved
- Sharp error bounds for the distribution of primes
- Hundreds of conditional theorems become unconditionally true
- Faster deterministic primality tests (proven so far only under the generalized hypothesis)
Open steps
- Capture all zeros, not a proportion Medium AI leverageReplace mollifier-type methods, which give only a proportion of zeros on the critical line (now about 67.2%), with an argument that reaches every zero.
- A Hilbert-Polya operator Low AI leverageConstruct a self-adjoint operator whose eigenvalues are the zeta zeros, giving the spectral explanation that random-matrix statistics hint at.
- Zero-free and zero-density bounds Medium AI leverageTighten explicit bounds on where exceptional zeros could lie (in the style of Guth and Maynard); many results that assume RH can then be made unconditional.
- High-precision zero computation Medium AI leverageVerify zeros and test pair-correlation statistics at much larger heights, as data for conjectures about the spectral structure.
Where AI could help
Medium AI leverage. AI just proved a record partial result (67.2% of zeros on the line), but a proof for all zeros needs an idea nobody has yet.
- Generate and test many proof strategies for mollifier and zero-density methods
- Run heavy symbolic and numerical computations on zero statistics and bounds
- Formalize new partial results in Lean so outside experts can check them quickly
- Search number theory and random-matrix literature for overlooked techniques
Shown so far
- In August 2026 Anthropic reported that an internal Claude research model proved that at least 67.2% of the non-trivial zeta zeros lie on the critical line (previous record 41.6%), with a Lean formalization, according to the company. source
Prerequisites
- Calculus (Analysis)1684
- Number Theory & Group Theory1801
- Prime Number Theorem1896The Riemann hypothesis would sharpen the prime number theorem