Riemann Hypothesis: all non-trivial zeros of ζ(s) lie on Re s = 1/2
Defining problem of analytic number theory with no credible complete line of attack, hence just below P vs NP (100) and above Yang–Mills (93) in difficulty. The 2026 proportion results and the claimed 7/8 half-plane are far from the line and unrefereed. Application value 30 (GRH-conditional algorithms and prime bounds would become unconditional) leads mathematics; leverage 95 is the highest after P vs NP because thousands of results are conditional on RH or GRH.
The problem
Do all the interesting zeros of the zeta function lie on one straight line? Nobody knows, and the answer controls the primes.
- 1Primes look irregular but average smoothly
- 2Zeta zeros add waves to the prime count
- 3Zeros on a single line mean tiny waves
- 4Nobody can prove all zeros stay on it
The statement. The Riemann zeta function , continued to the whole complex plane, has trivial zeros at and infinitely many non-trivial zeros in the critical strip . The (RH) says every non-trivial zero has . The generalized version says the same for every Dirichlet L-function and, more broadly, for L-functions of automorphic forms.
Built from
earlier problems → this oneThe problems this one is built from: progress or tools it needs (builds on), special cases that come first (generalizes), and weaker results on the way (implies). On the right, what builds on it.
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- Riemann Hypothesis: all non-trivial zeros of ζ(s) lie on Re s = 1/2This problem
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Harmonic analysisTheory
- Step 2
- Step 3
- The Riemann zeta function and prime distributionTheory · host
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- Riemann Hypothesis: all non-trivial zeros of ζ(s) lie on Re s = 1/2The problem
State of the art
Not proved. Unconditionally the best zero-free regions are of Vinogradov–Korobov type and shrink towards Re s = 1 as |t| grows. Hardy (1914) showed infinitely many zeros lie on the line; Selberg (1942) a positive proportion; Levinson (1974) more than 1/3; Conrey (1989) more than 2/5; Pratt, Robles, Zaharescu and Zeindler (2020) just over 5/12. In August 2026 Alpöge and Furman (arXiv:2608.13637, v1 13 August, unrefereed) proved that at least 2/3 of the zeros, counted with multiplicity, are simple and on the critical line (0.6725 with the Montgomery–Taylor window), with a Lean 4 formalization; the arXiv comments state that the proof was discovered autonomously by Claude (Anthropic) and verified and communicated by the authors. Lamzouri (arXiv:2609.02882, unrefereed) gave an independent, simpler proof of the 67.25% figure. Proportion results do not bear on the hypothesis itself: they say nothing about the remaining third of the zeros. OpenAI's 6 October 2026 release (github.com/openai/math, family 003) claims a zero-free half-plane Re s > 7/8 for ζ and all Dirichlet L-functions (a 'quasi-Riemann hypothesis'), with a second proof of Re s > 11/12 and a Lean formalization of the 7/8 statement listed in the repository; it is unrefereed and no independent assessment had been published by 15 October 2026. Even if correct, it leaves the strip 1/2 < Re s ≤ 7/8 open. Platt and Trudgian (2021) verified RH rigorously for |Im ρ| ≤ 3·10^12, and Gourdon (2004) checked about 10^13 zeros non-rigorously. Guth and Maynard (2024) lowered the zero-density exponent from Huxley's 12/5 to 30/13. The finite-field analogue is a theorem (Weil, Deligne), and the zeros show GUE statistics (Montgomery, Odlyzko), supporting a spectral (Hilbert–Pólya) picture that has not been realized.
Realistic applications
None directly. RH would sharpen error terms across prime number theory (primes in short intervals, least prime in progressions, deterministic primality and factoring-related bounds under GRH) and make many conditional results unconditional. Methods developed on the way, such as large-value estimates and decoupling, have been reused in harmonic analysis.
What counts as solved
A proof (or disproof by exhibiting a zero off the line) accepted by the analytic number theory community after refereeing, and ideally formalized. Numerical verification of finitely many zeros does not count. A zero-free region Re s > σ0 with σ0 < 1 would be a major partial result, not a resolution.
How it gets verified
Lean proofA formal proof checked by Lean 4 / Mathlib.
Statement: RiemannHypothesis.riemannHypothesis : RiemannHypothesis in Millennium/RiemannHypothesis.lean (Formal Conjectures pinned at commit e754a78 (17 October 2026)), where RiemannHypothesis is Mathlib's own definition in Mathlib.NumberTheory.LSeries.RiemannZeta: every zero of riemannZeta that is not a trivial zero (−2, −4, …) and not s = 1 has real part 1/2. A proof is accepted when it compiles against the pinned corpus without sorry and passes Comparator with only the standard axioms; a human-readable proof would still be refereed. Numerical verification of finitely many zeros does not count.
Claims under verification
+ add a claim- UnverifiedPostedProofOpenAI (model-produced manuscripts, github.com/openai/math family 003) · Oct 6, 2026Zero-free half-plane Re s > 7/8 for ζ and all Dirichlet L-functions0✓ 0✗ 4 open
- Partly verifiedPostedBoundLevent Alpöge and Ralph Furman (arXiv comments: proof discovered autonomously by Claude, verified and communicated by the authors) · Aug 13, 2026More than two thirds of the zeta zeros are simple and on the critical line
Where the work is
The missing ingredient is a new idea, not computation: numerics have verified 10^13 zeros without bearing on a proof. Formalization is a minor future share.
Bottlenecks
ranked by votes- #1TheoryNo operator or positivity mechanism forces Re ρ = 1/2
Hilbert–Pólya would need a self-adjoint operator whose spectrum is the zero set, and Weil positivity needs an analogue of the Hodge index theorem on 'Spec ℤ'. Neither the operator nor the arithmetic surface is known, and the function-field proof uses geometry that has no number field counterpart.
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Connected problems
Progress
- Aug 13, 2026At least 2/3 of zeros simple and on the line (unrefereed)source
Alpöge–Furman (v1 13 August 2026), Lean-formalized; Lamzouri gives an independent proof in September.
- May 31, 2024Guth–Maynard zero-density exponent 30/13source
First improvement of Huxley's exponent 12/5 in four decades.
- Apr 21, 2020Rigorous check up to height 3·10^12source
Platt and Trudgian verify RH to |Im ρ| ≤ 3·10^12 with interval arithmetic.
- Nov 1, 1859Riemann states the hypothesissource
In 'Über die Anzahl der Primzahlen unter einer gegebenen Grösse' (date approximate to the month).