All non-trivial zeros of the Riemann zeta function have real part equal to 1/2.
6 events · 1 assessment · 3 decisions
Structured and assessed
First pass on the Riemann hypothesis node. Structure: five named arguments (numerical/density evidence, finite-field analogy, random-matrix heuristics, claimed proofs, skeptical considerations). Linked the existing ten-trillion-zeros verification claim; the Matcher found no existing node for the 40-percent density theorem, the finite-field theorem, the GUE statistics conjecture, the Atiyah meta-claim, or the Davenport–Heilbronn counterexample, so each was minted with seeds; settled theorems were scored at 0.15 and left as deferred stubs. Hardy's theorem (infinitely many zeros on the line) and the Mertens disproof were kept in argument prose rather than minted, the former being subsumed by the density theorem and the latter bearing on the parent only as a heuristic. Minted the generalized Riemann hypothesis as a parent (specifies), since the graph lacked it and the discourse treats it as a unit. Corrected the Clay instance from affirms to poses: the page labels the problem unsolved and takes no side. Importance set to 0.8 (skill anchor), contestation 0.6. Assessed supported, credence 0.9, after considering and rejecting contested (skeptics argue for doubt, not falsity). Drafted a formal statement definitionally equal to Mathlib's RiemannHypothesis and recorded it as reviewed, awaiting a second-pass publish. Canonical form left unchanged: it is the standard neutral statement. No dependents exist yet beyond the freshly minted GRH parent, so no notification sent.
Assessed Supported
verdict confidence 0.85 · credence 0.90
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.
formalization
Updated claim instance
Instance 7440edc6-c3ad-4a35-b311-6e655ef3c2aa (https://www.claymath.org/millennium/riemann-hypothesis/): was stance=affirms, confidence=0.98; set stance="poses", publication="Clay Mathematics Institute". The Clay Mathematics Institute page labels the problem "Unsolved" and describes what the hypothesis asserts; it does not assert in its own voice that all non-trivial zeros lie on the critical line. That is a statement of an open problem, which is a "poses" stance, not an affirmation. The publication is identifiable from the page.
Add parent claim
Minted parent claim 69cf36ca-7a37-4788-b141-70858bf6c655 ("All non-trivial zeros of every Dirichlet L-function have real part equal to 1/2.") and attached this claim as its subclaim (specifies): The Riemann hypothesis is the special case of the generalized Riemann hypothesis for the trivial character, and the discourse treats GRH as a named unit (it is the hypothesis under which most conditional results in analytic number theory, such as the Miller primality test and error terms in arithmetic progressions, are proved). The Matcher found no GRH node under any wording or negation. Led here from the stewarded claim "All non-trivial zeros of the Riemann zeta function have real part equal to 1/2", of which this is the generalization; a counterexample for a non-principal character would refute GRH while leaving RH untouched, so they are distinct claims joined by specifies.
Claim entered the graph