{ "artifactId": "artifact.library.mathlib.basel-problem.v001", "candidateOnly": false, "category": "Number theory", "claimBoundary": "This page presents Mathlib's real HasSum statement for n ↦ 1/(n:ℝ)^2 with value π^2/6. Its natural-number indexing includes n = 0, whose term is zero under Lean's totalized real division, so it represents the usual series over positive integers. The selected declaration is only the exponent-two specialization, not a generic theorem about all zeta values.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem hasSum_zeta_two : HasSum (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 2) (π ^ 2 / 6)", "family": "Special values of zeta series", "id": "library.mathlib.basel-problem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/basel-problem/", "entryData": "/data/library-theorems/basel-problem.json", "evidence": "/proofs/artifact.library.mathlib.basel-problem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.basel-problem.v001.evidence.json", "source": "/sources/upstream/basel-problem/" }, "landmark": { "completeFormalType": "theorem hasSum_zeta_two : HasSum (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 2) (π ^ 2 / 6)", "concepts": [ "Basel problem", "reciprocal-square series", "Bernoulli functions", "Fourier coefficients", "even zeta values" ], "plainLanguage": "The sum of the reciprocals of the positive perfect squares is exactly π squared divided by six.", "poster": { "alt": "A dark green and ivory theorem poster states the reciprocal-square identity and traces the checked route from Bernoulli functions through Fourier coefficients and the general even-power sum to exponent two.", "byteSize": 2305542, "caption": "Mathlib's exponent-two specialization turns the Bernoulli–Fourier evaluation into the exact reciprocal-square sum.", "description": "The familiar reciprocal-square equation dominates a sparse engraved poster. A periodic Bernoulli ribbon, harmonics with square-diminishing amplitudes, and a six-bay circle-and-square seal support four short source movements and an exact-scope footer explaining the natural-index zero term.", "derivatives": [ { "byteSize": 102450, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/basel-problem/theorem-poster-v1-640.webp", "sha256": "sha256:5371f8f48e37f82efa67167c03c8214c8c53cdc32004d3d50cb6a4f7c9373b1c", "width": 640 }, { "byteSize": 198658, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/basel-problem/theorem-poster-v1-1024.webp", "sha256": "sha256:3facab747d2a2a6e5aaad80434cfe3ceb785768ea524a99c84fb45030fa9ef7f", "width": 1024 } ], "generatedAt": "2026-07-25T16:36:07-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/basel-problem/theorem-poster-v1.png", "sha256": "sha256:2e8beef1b8407a53f8ce316ead5f0c215181009207bf289b08cb770a7ab6bfd5", "title": "Basel Problem — reciprocal squares sum to π²/6", "width": 1024, "kind": "theorem_poster", "transcript": "NUMBER THEORY · RECIPROCAL SQUARES\nTHE BASEL PROBLEM\n1 + 1/2² + 1/3² + ··· = π²/6\nRECIPROCAL SQUARES CONVERGE TO A CIRCLE CONSTANT\nHOW THE CHECKED ROUTE MOVES\n1 · Build Bernoulli functions\n2 · Compute Fourier coefficients\n3 · Obtain the general even-power sum\n4 · Specialize to exponent two\nEXACT SCOPE\nLean starts at n = 0; that term is zero. This is exponent two only." }, "visual": { "alt": "On a dark engraved field, a periodic Bernoulli ribbon resolves into symmetric Fourier waves whose square-diminishing gold tiles accumulate toward a six-bay circle-and-square seal.", "byteSize": 2905378, "caption": "The reciprocal-square coefficients converge through the Bernoulli–Fourier route to π²/6.", "description": "A single exponent-two lane links a periodic Bernoulli ribbon to successively finer Fourier waves. Gold square tiles diminish along the harmonic strands and accumulate at a precise circle-and-square seal divided into six equal bays, while a separate null starting socket marks the zero natural-index term.", "derivatives": [ { "byteSize": 51098, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/basel-problem/theorem-schematic-v1-640.webp", "sha256": "sha256:94f1f7ec7d1057242d81e07526a1155a894017b82b12ed52fe116dc29c9034c1", "width": 640 }, { "byteSize": 196812, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/basel-problem/theorem-schematic-v1-1200.webp", "sha256": "sha256:bfd65bea9dfe01ec0ee9fb5ccc9f831dfd0607c21b730956438109cd22d1be28", "width": 1200 } ], "generatedAt": "2026-07-25T16:36:07-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/basel-problem/theorem-schematic-v1.png", "sha256": "sha256:0d1a13440cce85c1251a0a70a1f465fb1866b42133b506675e1807083b0bdf88", "title": "Reciprocal-square harmonics converge to a circle constant", "width": 1536 }, "walkthrough": null, "whyLandmark": "The Basel problem is one of the most recognizable exact evaluations of an infinite series. Mathlib's declaration exposes it as a concise specialization resting on a substantial Bernoulli-function and Fourier-coefficient development." }, "nonClaims": [ "The exact declaration is a real HasSum statement indexed by natural numbers.", "The n = 0 term is zero because real division in Lean is totalized; the nonzero terms form the usual positive-index series.", "The selected declaration concerns exponent two only and is not itself the general even-zeta theorem.", "It gives an exact sum, not a quantitative convergence rate or error bound.", "ProofAtlas is indexing an existing Mathlib theorem, not claiming a new proof or new mathematics.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.basel-problem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "The reciprocal-square series has sum π²/6.", "targetId": "target.library.mathlib.basel-problem.v001", "title": "Basel Problem", "upstreamOrigin": { "declarationName": "hasSum_zeta_two", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/NumberTheory/ZetaValues.lean#L456", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:2db7a16478fd0d1560592e5bdaeffc6b1dc4d2f916037f15def076311748def1", "sourceByteLength": 21906, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/NumberTheory/ZetaValues.lean", "sourceLine": 456, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }