Analytic number theory · complex analysis · zero geometry

Riemann Hypothesis

Collaboration beta

Do every nontrivial zero of the completed zeta function lie on the critical line with real part one half?

ξ (ρ) = 0 \Rightarrow Re ρ = \frac{1}{2}

Clay Millennium Prize ProblemHilbert's Eighth Problem — Riemann-hypothesis component

Known results and sources

The critical line sits exactly midway across the strip where the Riemann Hypothesis predicts every nontrivial zero must lie.

Research problem

Exact mathematical statement

Define the completed zeta function

\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).

The Riemann Hypothesis asks whether

\xi(\rho)=0\quad\Longrightarrow\quad\operatorname{Re}\rho=\frac12.

Centering at one half gives the even entire function

\mathcal X(w)=\frac{\xi(1/2+w)}{\xi(1/2)},

so the same question asks whether every zero of $\mathcal X$ lies on $\operatorname{Re}w=0$ . The conjecture remains open; this page follows one unfinished analytic route rather than presenting a proof.

Problem infographic

Problem at a glance

Scientific explainer for the open Riemann Hypothesis, showing the completed zeta function, the critical strip zero-location question, the critical line Re(s)=one half, and the equivalent centered statement that every zero of the even entire function X lies on Re(w)=0. — The completed zeta function, its critical strip, and the centered imaginary-axis formulation of the still-open zero-location question.

Current mathematical picture

Where work on Riemann Hypothesis stands

Open problem

Revision 6 develops an analytic program around the completed zeta function. It derives a reflection quotient, obtains the zero-freeness needed for a global principal logarithm, develops a directed nodal graph with strict phase flow, classifies local singularities and possible component ends, excludes several endpoint and compact-component scenarios, gives an analytic off-line zero-free band through height 39/10, and narrows several failed routes. The leading unresolved gate is global endpoint incidence and same-sign phase pairing, together with exclusion of zero-phase singular vertices. These arguments remain provisional and do not prove or disprove the Riemann Hypothesis.

Strongest supported footholdShift-two positivity and zero-free transform

The current work reports shift positivity through two, zero-freeness of L on Re(z)>=-2, the global logarithmic strip, and sharp Stieltjes moment reductions.

Evidence posture · Reported result

Leading routeDirected nodal graph and endpoint pairing

Leading route: use the source-reported proper analytic nodal graph, strict phase flow, singular normal forms, and classified ends to determine global incidence and prove component-wise phase avoidance.

Route status · Active route

Useful failureHeight-growing Pick/Padé hierarchy

Narrowed backup: interpolation order must grow with height and provide explicit complex-domain error below the completed-xi scale together with exact conditioning control.

Route status · Narrowed route

Main reductionProper components and classified ends

The current work reports that target nodal components are proper, have no bottom, origin, or compact-component endpoints, and can end only at imaginary-boundary portals, right-edge intersections, or infinity; regular infinity edges are phase-safe.

Evidence posture · Source-reported route statement

Completed special caseAnalytic zero-free band through 39/10

The current work reports a Pick-based analytic off-line exclusion through absolute height 39/10 without relying on a finite-height numerical zero certificate.

Evidence posture · Reported special case

Priority open bridgeProve directed same-sign endpoint pairing

Prove that no connected component of the target nodal graph has a phase image containing zero.

Task status · Prerequisites still open

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Riemann Hypothesis in numbers

3.6kretained lines of mathematical investigation3,641 in the current working snapshot

Argument development: 3,200 · 88%
Explored or eliminated routes: 47 · 1%
Computational analysis: 108 · 3%
Open obligations: 125 · 3%
Definitions and setup: 161 · 4%

12selected mapped statements5routes investigated5reported milestones5open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byBernhard RiemannProposed1859Status checkedAug 4, 2026

Source recordedAug 4, 2026

Recommended next task

Build the global endpoint-incidence atlas

Determine which portal, right-edge, singular-vertex, top-cut, and infinity ends belong to each global component of the proper directed nodal graph.

Suggested move: Use growing generic rectangles, label every truncated edge endpoint, pass through singular vertices with the alternating flow rule, and give a locally finite direct-limit argument.

What would count as progress

Every permitted endpoint and infinity end has a multiplicity-safe component-incidence assignment.
Any infinite family of vertices or ends is controlled by a proved locally finite direct limit.
The first exact ambiguity is returned if a complete pairing cannot be proved.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

23 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeDirected nodal graph and endpoint pairing

Route status · Active route

Active routeEndpoint phases and rectangle flow balance

Active companion route: determine portal and right-edge phase signs and combine them with multiplicity-safe rectangle accounting to forbid opposite-sign pairing.

Route status · Active route

Explored alternatives

Other routes

3 recorded

Narrowed routeHeight-growing Pick/Padé hierarchy

Narrowed backup: interpolation order must grow with height and provide explicit complex-domain error below the completed-xi scale together with exact conditioning control.

Route status · Narrowed route

Narrowed routeTheta-density localization

Narrowed backup: generic moment positivity is sharp, so a closure must use a quantitative property of the actual theta density that excludes the forced zero data.

Route status · Narrowed route

Route held in reserveFourier-decay and scalar-growth closures

Retained but not prioritized over the directed reflection graph: almost-pi/2 reflection-profile decay or the scalar subexponential bound would provide equivalent closure routes.

Route status · Route held in reserve

Route statements and reductions

Statements the next route can inspect and build on

Route statementGlobal endpoint-incidence and same-sign pairing gate

Every connected component of the target nodal set has phase image disjoint from zero.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks

Build the global endpoint-incidence atlas

Determine which portal, right-edge, singular-vertex, top-cut, and infinity ends belong to each global component of the proper directed nodal graph.

Ready to work on

Determine portal and right-edge phase signs

Obtain theta-specific phase-sign and incidence information at imaginary-boundary portals and right-edge intersections.

Suggested move: Retain the theta lattice before absolute values and focus signed Stokes analysis on incidence and finite endpoint signs rather than re-proving safety of an already identified escaping edge.

Ready to work on

Test an exponentially resolving or theta-density closure

Either quantify a height-growing Pick/Padé hierarchy below the completed-xi scale or prove a theta-density inequality excluding the exact forced moments.

Suggested move: For interpolation, quantify m(gamma), feasible-region diameter, and conditioning; for density, isolate a genuine property of the explicit theta density rather than generic positivity.

Ready to work on

Prove directed same-sign endpoint pairing

Prove that no connected component of the target nodal graph has a phase image containing zero.

Suggested move: Record portal and right-edge phase signs and combine the directed graph with a planar-flow, argument-principle, or multiplicity-safe rectangle-winding identity.

Prerequisites still open

Exclude zero-phase singular vertices

Eliminate the simultaneous target-strip system U=0, V=0, and h'=0, or produce an exact solution.

Suggested move: Test whether the two exact L-equations force an impossible positive-measure covariance identity, while keeping every Stieltjes/Pick input within its stated scope.

Prerequisites still open

Sourced mathematical context

The known mathematical landscape

Context collected Aug 4, 2026

Current statusOpen problem

The classical Riemann Hypothesis remains open: every nontrivial zero of the Riemann zeta function is conjectured to have real part 1/2. Rigorous computation verifies the claim only through finite height 3×10^12, and unconditional theory places slightly more than five-twelfths of the zeros on the critical line. Neither result extends to every zero. No reviewed proof or counterexample currently changes this status.

[3][5][8]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2021
Computational resultPlatt and Trudgian rigorously verified with interval arithmetic that every zero through height 3×10^12 lies on the critical line and is simple. A finite-height result cannot settle the all-heights hypothesis.[8]
2020
Peer reviewedRodgers and Tao proved that the de Bruijn–Newman constant is nonnegative. Since the Riemann Hypothesis is equivalent to the complementary inequality Λ≤0, it is now equivalent to Λ=0.[7]
2019
Peer reviewedPratt, Robles, Zaharescu, and Zeindler proved unconditionally that more than five-twelfths of the nontrivial zeros lie on the critical line.[6]
1914
Authoritative summaryHardy proved that infinitely many nontrivial zeros lie on the critical line. This establishes infinitely many instances, not that every nontrivial zero lies there.[4]

Mathematical neighborhood

Related results and reusable starting points

Current focusRiemann Hypothesis

Equivalent formulationsquare-root-scale error in the prime number theorem

The hypothesis is equivalent to the near-optimal prime-counting error estimate π(x)=Li(x)+O(sqrt(x) log x).

[4]

Equivalent formulationLagarias divisor-sum and harmonic-number criterion

Lagarias gave an elementary equivalent involving the divisor-sum function and harmonic numbers, converting the analytic zero-location claim into inequalities for every positive integer.

[9]

Equivalent formulationLi positivity criterion

Li's criterion reformulates the hypothesis as positivity of an infinite sequence of coefficients derived from logarithmic derivatives of the completed zeta function.

[10]

Equivalent formulationde Bruijn–Newman constant Λ=0

After the nonnegative lower bound for the de Bruijn–Newman constant, the Riemann Hypothesis is equivalent to the exact equality Λ=0.

[7]

Stronger or generalized formgeneralized Riemann hypotheses

Generalized Riemann hypotheses extend critical-line predictions to broader families of Dirichlet, Dedekind, automorphic, and other L-functions. They must not be conflated with the classical statement.

[4]

Related problemRiemann hypotheses over finite fields

The corresponding Riemann hypotheses for zeta functions of varieties over finite fields were proved through the work of Weil and Deligne. These geometric analogues strongly influence strategy but do not prove the classical hypothesis.

[4]

Related problemHilbert–Pólya spectral program

The Hilbert–Pólya program seeks a self-adjoint operator whose spectral data encode zeta zeros; such an operator with the needed properties would force the relevant spectral parameters to be real.

[4]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal statement · statement onlyMathlib.RiemannHypothesis
Mathlib defines the classical Riemann Hypothesis as a Lean proposition: a nontrivial zero of riemannZeta, excluding the pole at 1, has real part 1/2. The declaration is a statement, not a proof.
[11]
formal library support · source linked; not reproduced by ProofAtlasMathlib Riemann zeta and zero-set infrastructure
Mathlib develops the Riemann zeta function, completed zeta constructions, functional equations, analytic properties, and a discrete closed set of zeta zeros. This is reusable infrastructure for formal attempts but does not close the hypothesis.
[11][12]
computation · not independently reproducedPlatt–Trudgian rigorous finite-height verification
The peer-reviewed interval-arithmetic computation verifies the hypothesis and simplicity of the zeros through height 3×10^12. ProofAtlas has not independently rerun it, and the finite computation does not establish the infinite theorem.
[8]
dataset · source linked; not reproduced by ProofAtlasLMFDB Riemann zeta zeros dataset
LMFDB links a 1.32 TB auxiliary dataset containing the first 10^11 Riemann zeta zeros. It is a computational resource for experiments and checking finite phenomena, not theorem evidence for all zeros.
[13]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA checked proof term for Mathlib.RiemannHypothesis is not present in the verified public library material.
Formalization targetAny selected packet route must be translated into exact Lean definitions and connected by statement-aligned lemmas to Mathlib.RiemannHypothesis.
Formalization targetRoute-specific advanced analytic machinery may still be absent even where Mathlib supplies the zeta function and its canonical statement.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected supporting details in the research record. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

Directed nodal graph research baselineRevision 6 retains a substantial analytic program and isolates global endpoint incidence, same-sign phase pairing, and zero-phase singular exclusion as the leading unresolved gate.

Mapped research milestoneInitial research sequence

Analytic route assembled

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

9 standing statements3 proposed statements5 mathematical milestones5 open questions2 narrowed routes1 conditional results1 completed special cases

Statements by mathematical role12 selected mapped statements

theorem candidate2 of 122
equivalence1 of 121
lemma6 of 126
reduction2 of 122
negative result1 of 121

Selected mathematical clusters5 mathematical clusters

Exact zero-location formulationsThe completed-zeta statement, centered entire function, and reflection-quotient phase gate.2 displayed rows

retained route statementRiemann Hypothesis
retained route statementReflection-quotient phase gate

One-sided transform and logarithmic stripSource-reported shifted sine positivity, zero-free transform, and global logarithm.4 displayed rows

retained route statementShifted sine positivity through shift twointermediate
retained route statementOne-sided transform zero-free half-planeintermediate
retained route statementGlobal principal logarithmintermediate
retained route statementAnalytic off-line zero-free bandspecial case

Directed nodal graph frontierLocal phase geometry and classified ends reduce the leading program to global incidence, same-sign pairing, and singular exclusion.11 displayed rows · 2 routes included

retained route statementStrict phase flow on regular nodal edgesintermediate
retained route statementExact local singular normal formintermediate
retained route statementProper components and classified endsintermediate
retained route statementGlobal endpoint-incidence and same-sign pairing gate
retained route statementZero-phase singular exclusion
Research targetBuild the global endpoint-incidence atlasopen
Research targetProve directed same-sign endpoint pairingopen
Research targetExclude zero-phase singular verticesopen
Research targetDetermine portal and right-edge phase signsopen
Active routeDirected nodal graph and endpoint pairingLeading route: use the source-reported proper analytic nodal graph, strict phase flow, singular normal forms, and classified ends to determine global incidence and prove component-wise phase avoidance.
Active routeEndpoint phases and rectangle flow balanceActive companion route: determine portal and right-edge phase signs and combine them with multiplicity-safe rectangle accounting to forbid opposite-sign pairing.

Narrowed alternative closuresHeight-growing interpolation, theta-density localization, and retained Fourier/scalar backups after scoped route eliminations.8 displayed rows · 3 routes included

retained route statementFixed-distinct-node interpolation blindnessconditional
Useful failureAbstract Stieltjes positivity without theta-density structurereported failure
Useful failureFixed finite distinct-node Pick interpolation with polynomial marginreported failure
Useful failurePositive Stieltjes shifts growing proportionally with heightreported failure
Research targetTest an exponentially resolving or theta-density closureopen
Narrowed routeHeight-growing Pick/Padé hierarchyNarrowed backup: interpolation order must grow with height and provide explicit complex-domain error below the completed-xi scale together with exact conditioning control.
Narrowed routeTheta-density localizationNarrowed backup: generic moment positivity is sharp, so a closure must use a quantitative property of the actual theta density that excludes the forced zero data.
Route held in reserveFourier-decay and scalar-growth closuresRetained but not prioritized over the directed reflection graph: almost-pi/2 reflection-profile decay or the scalar subexponential bound would provide equivalent closure routes.

Evidence and trust boundaryRecorded packet audit transcripts and finite-height candidates remain distinct from mathematical acceptance.2 displayed rows

ComputationBundled Revision 5 exact symbolic/rational audit, Revision 6 structural audit, Python compilation transcript, and theta-kernel monotonicity checker transcript.The retained transcript reports that all listed checks passed. ProofAtlas did not execute the bundled scripts, so the result remains source-reported rather than independently reproduced here. · reported unreproduced
ComputationInherited finite-height exclusion programs at heights 10, 15.53, and 50.The current work reports finite-height exclusion candidates, but directed-rounding and implementation seams remain unresolved and no candidate is promoted to theorem status. · reported unreproduced

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve that no connected component of the target nodal graph has a phase image containing zero.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Every finite endpoint and singular vertex is assigned a compatible phase sign.
Opposite-sign endpoints are excluded from each component.
Infinity phase zero occurs only as a nonattained endpoint value.

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Prepared starting pointBuild the global endpoint-incidence atlas

Riemann Hypothesis · ready to start

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Research contextPrepared context for any AI agent

Do every nontrivial zero of the completed zeta function lie on the critical line with real part one half?

Exact question and boundaries
Current routes and known obstacles
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Sources and references15 cited works · next context review by Nov 4, 2026

The mathematical context was checked on Aug 4, 2026. Status can be refreshed sooner after a material result or claim.

1
Ueber die Anzahl der Primzahlen unter einer gegebenen Grösseoriginal source · Bernhard Riemann · Monatsberichte der Berliner Akademie · 1859 · accessed Aug 4, 2026
2
Riemann's 1859 Manuscriptauthoritative webpage · Clay Mathematics Institute · accessed Aug 4, 2026
3
Riemann Hypothesis — Millennium Prize Problemmaintained problem list · Clay Mathematics Institute · accessed Aug 4, 2026
4
Problems of the Millennium: The Riemann Hypothesissurvey or monograph · Enrico Bombieri · Clay Mathematics Institute · 2000 · accessed Aug 4, 2026
5
DLMF §25.10: Zeros of the Riemann Zeta Functionencyclopedia · National Institute of Standards and Technology · accessed Aug 4, 2026
6
More than five-twelfths of the zeros of ζ are on the critical linepeer reviewed result · Kyle Pratt, Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler · Research in the Mathematical Sciences · 2019-12-06 · ARXIV 1802.10521 · DOI 10.1007/s40687-019-0199-8 · accessed Aug 4, 2026
7
The de Bruijn–Newman constant is non-negativepeer reviewed result · Brad Rodgers, Terence Tao · Forum of Mathematics, Pi · 2020 · DOI 10.1017/fmp.2020.6 · accessed Aug 4, 2026
8
The Riemann hypothesis is true up to 3×10^12peer reviewed result · Dave Platt, Tim Trudgian · Bulletin of the London Mathematical Society · 2021 · ARXIV 2004.09765 · DOI 10.1112/blms.12460 · accessed Aug 4, 2026
9
An Elementary Problem Equivalent to the Riemann Hypothesispeer reviewed result · Jeffrey C. Lagarias · The American Mathematical Monthly · 2002 · ARXIV math/0008177 · DOI 10.1080/00029890.2002.11919883 · accessed Aug 4, 2026
10
The Positivity of a Sequence of Numbers and the Riemann Hypothesispeer reviewed result · Xian-Jin Li · Journal of Number Theory · 1997 · DOI 10.1006/jnth.1997.2137 · accessed Aug 4, 2026
11
Mathlib.NumberTheory.LSeries.RiemannZetaformalization · Mathlib · accessed Aug 4, 2026
12
Mathlib.NumberTheory.LSeries.ZetaZerosformalization · Mathlib · accessed Aug 4, 2026
13
LMFDB Auxiliary Datasets — Zeros of ζ(s)software or dataset · LMFDB Collaboration · accessed Aug 4, 2026
14
Hilbert problems — Hilbert's eighth problemencyclopedia · Encyclopedia of Mathematics · accessed Aug 4, 2026
15
Riemann hypothesisencyclopedia · Wikimedia Foundation · accessed Aug 4, 2026

Important qualifications

This record describes the classical Riemann Hypothesis for the Riemann zeta function, not the generalized Riemann hypothesis for Dirichlet, Dedekind, automorphic, or other L-functions.
No exact problem-level Riemann Hypothesis entry was verified in the current Epoch FrontierMath Open Problems collection; related problems or problems conditional on a generalized Riemann hypothesis do not establish membership.
The scoped formalization review verified a canonical Lean statement and substantial zeta-function support in Mathlib, but no checked proof of the Riemann Hypothesis.
Mathlib is a moving library. Any formal-library declaration shown publicly should be pinned to an exact source revision.
The LMFDB auxiliary dataset page advertises the first 10^11 zeta zeros, while its associated explanatory knowledge page is marked awaiting review; the data are useful computational material, not a proof of the infinite statement.
The Platt–Trudgian finite-height verification was not independently rerun by ProofAtlas during this administrative collection.
Recent manuscripts and purported proofs do not change the problem's open status without authoritative mathematical acceptance.

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