All non-trivial zeros of the Riemann zeta function have real part equal to 1/2.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
The Riemann hypothesis, stated by Riemann in 1859, remains unproved and undisproved. It is Hilbert's eighth problem and one of the seven Millennium Prize Problems, and no claimed proof, including the argument announced by Michael Atiyah in 2018, has been accepted by the mathematical community. Its status is therefore that of an open conjecture, but one carrying evidence that mathematicians count for a great deal.
That evidence takes several forms. Rigorous computation has confirmed that the first ten trillion or so non-trivial zeros lie on the critical line, most recently Platt and Trudgian's interval-arithmetic verification to height 3·10^12 (2021). It is a theorem that at least 40 percent of all non-trivial zeros lie on the critical line, following Selberg, Levinson and Conrey, and zero-density estimates, sharpened in 2024 by Guth and Maynard, bound how many zeros could lie off it. Structurally, the analogue of the hypothesis for zeta functions over finite fields is a proven theorem (Weil, Deligne), which suggests the classical statement is true for reasons a proof over the rationals has yet to capture. And the zeros' statistics agree, as far as they have been computed, with the eigenvalue statistics of random Hermitian matrices, consistent with the Hilbert–Pólya picture of the zeros as a self-adjoint spectrum.
A credible minority of analysts urge caution. The functional equation alone does not force the hypothesis, as the Davenport–Heilbronn function shows; the Mertens conjecture, which would have implied the hypothesis and had extensive numerical support, was disproved in 1985; and the computed zeros are a vanishingly small initial segment in a problem where the relevant scale grows like the logarithm of the logarithm of the height. Aleksandar Ivić collected these reasons for doubt; David Farmer has argued in reply that none of them survives closer analysis. The disagreement is empirical in character and would be resolved by a proof, by a disproof, or by a zero found off the line.
Full reasoning: the evidence and decisions behind this verdict
The proposition is open: no proof or disproof exists, so under the mathematics domain's statuses the choice is between "unsupported" (an open conjecture with nothing beyond plausibility) and "supported" (an open claim with evidence mathematicians count). The evidence here is substantial and of several independent kinds, so "supported" is the right reading.
For: (1) Verification for the first ten trillion zeros: Gourdon's 2004 computation reached 10^13 zeros; the refereed rigorous record is Platt and Trudgian, Bull. LMS 53 (2021), verifying the hypothesis to height 3·10^12 with interval arithmetic (arxiv.org/abs/2004.09765). (2) At least 40 percent of zeros on the line: Conrey 1989, improved to about 41.7 percent by Pratt, Robles, Zaharescu and Zeindler (2020); these are theorems. Guth and Maynard's 2024 large-value estimates for Dirichlet polynomials give the zero-density bound N(σ,T) ≤ T^{30(1−σ)/13+o(1)}, the first improvement on Ingham's 1940 exponent (arxiv.org/abs/2405.20552); this bounds possible exceptions rather than excluding them. (3) The finite-field analogue is a theorem (Weil 1940s for curves, Deligne 1974 in general). (4) GUE statistics of the zeros: Montgomery's pair-correlation theorem (conditional, restricted test functions) and Odlyzko's computations near the 10^20th zero. Each of these is evidence mathematicians in the field explicitly count toward the hypothesis.
Against: the Davenport–Heilbronn function shows a Riemann-type functional equation without an Euler product does not force zeros onto the line; the Mertens conjecture, numerically well supported and implying the hypothesis, is false (Odlyzko and te Riele 1985); Lehmer's phenomenon of nearly coincident zeros shows the function comes close to violating the hypothesis; and the scale on which the zeta function's behaviour changes grows like log log T, so computation to height 10^12 explores very little. Ivić (arxiv.org/abs/math/0311162) lays these out; Blanc (arxiv.org/abs/1706.09740) adds a heuristic locating possible counterexamples near unusually large peaks of |ζ|. Farmer (Bull. AMS, arxiv.org/abs/2211.11671) examines each published reason for doubt and argues, using random-matrix models and theorems on carrier waves, that none gives a good reason to doubt the hypothesis while agreeing that the computational evidence by itself is misleading. Neither side claims a decisive argument; the doubt is about how much weight the heuristics bear.
Claimed proofs: Atiyah's 2018 argument rests on a "Todd function" whose stated properties appear inconsistent, was never published, and is not accepted; no other announced proof has passed review. The Clay Mathematics Institute continues to list the problem as unsolved. The one recorded instance, the Clay problem page, poses the hypothesis without taking a side, so the instance set carries no stance signal and the verdict rests on the literature.
Credence 0.9 reflects near-unanimous expert expectation that the hypothesis is true, discounted for the historical record of numerically supported conjectures in this area failing at large heights (Mertens, Skewes-type sign changes) and for the absence of any structural argument over the rationals comparable to Deligne's. Verdict confidence 0.85: "supported" is clearly the right status among the open-problem statuses; the alternative reading, "contested" on the strength of the skeptical literature, was considered and rejected because the skeptics argue for doubt rather than for falsity and none denies the proposition. What would change the verdict: an accepted or machine-checked proof (verified), a zero off the line or a proof of the negation (contradicted), or a shift in expert opinion toward genuine disbelief (contested).
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because every one of the first ten trillion non-trivial zeros has been computed to lie on the critical line, and because it is a theorem that at least two fifths of all non-trivial zeros lie on the critical line, the zeros examined so far and a proven positive share of all zeros behave exactly as the hypothesis predicts, which raises confidence that every zero does.
The premises are established: the computed zeros all lie on the line and a proven two fifths of all zeros do. The inference is inductive rather than deductive: a finite initial segment and a proper fraction of the zeros cannot establish a universal statement, and the history of the Mertens conjecture shows that numerical support of this kind can fail at heights beyond computation. The argument raises confidence substantially without settling the question.
Because the analogue of the hypothesis for zeta functions of varieties over finite fields is a proven theorem (Weil for curves, Deligne in general), and the classical zeta function sits in the same family of L-functions with an Euler product and a functional equation, the classical statement is expected to hold for the same kind of structural reason, even though no analogue of the cohomological proof is known over the rationals.
The premise, that the finite-field analogue is a theorem, is settled. The inference is by analogy: the cohomological structure Deligne's proof relies on has no known counterpart over the rationals, so the argument shows the hypothesis is the kind of statement that is true in every setting where it can be proved, not that it is true in this one. It is the principal structural reason experts expect the hypothesis to hold.
Because the spacing statistics of the zeros match those of eigenvalues of random Hermitian matrices, the zeros look like the spectrum of a self-adjoint operator, and given the Hilbert–Pólya heuristic that such an operator would have real eigenvalues corresponding to zeros on the critical line, the statistical agreement is read as indirect evidence that the hypothesis is true.
The argument stands or falls with the agreement between zero statistics and random-matrix eigenvalue statistics, which is a conjecture with strong numerical and partial theoretical support rather than a theorem, and with the Hilbert–Pólya heuristic that such statistics indicate a self-adjoint spectrum, for which no operator has been found. Even granting both, the inference is heuristic: eigenvalue-like statistics are consistent with the hypothesis but do not entail it.
If Michael Atiyah's 2018 claimed proof establishes the Riemann hypothesis, the claim is a theorem; the same holds for any of the many other announced proofs, none of which has been accepted by referees or the number-theory community.
The inference is valid: a correct proof would make the claim a theorem. The argument lives or dies on whether Atiyah's 2018 argument establishes the hypothesis, and the mathematical community's examination found that it does not, so the argument currently carries no weight.
Because the Davenport–Heilbronn function satisfies a Riemann-type functional equation yet has zeros off the critical line, the functional equation alone cannot force the zeros onto the line, so any argument for the hypothesis must use the Euler product in an essential way; and because the Mertens conjecture, which had extensive numerical support and would have implied the hypothesis, was disproved by Odlyzko and te Riele in 1985, while the computed zeros form a vanishingly small initial segment in which the Lehmer phenomenon of near-coincident zeros already appears, some analysts hold that the numerical evidence is weaker than it looks and that the hypothesis could fail at heights beyond computational reach.
Its factual premises are settled: the Davenport–Heilbronn function has zeros off the line and the Mertens conjecture is false. What they establish is limited: that the functional equation alone cannot prove the hypothesis and that numerical evidence in this area is less conclusive than it appears. They are reasons to withhold certainty, not evidence that any zeta zero lies off the line, and Farmer's reply argues that random-matrix models account for each phenomenon without threatening the hypothesis.
Provenance
Where this claim has been said, linked to its canonical form.
it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.
Formulated in Riemann's 1859 paper, describing the Riemann hypothesis.
Asserted without evidence of the source's own. The Clay Mathematics Institute page states what the hypothesis asserts and marks the problem unsolved; it takes no position on its truth and offers the ten-trillion-zero computation only as context.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 13, 2026. Every judgment on this page is accompanied by a reasoning trace.