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It is well known that the $k$-th Dirichlet eigenvalue $\\lambda_k$ obeys the Weyl asymptotic formula, that is,\n\\[\n\\lambda_k\\sim\\frac{4\\pi^2}{(\\omega_n\\mathrm{vol}\\Omega)^\\frac{2}{n}}k^\\frac{2}{n}\\qquad\\hbox{as}\\quad k\\rightarrow\\infty,\n\\]\nwhere $\\mathrm{vol}\\Omega$ is the volume of $\\Omega$. In view of the above formula, P\\'{o}lya conjectured that\n\\[\n\\lambda_k\\gs\\frac{4\\pi^2}{(\\omega_n\\mathrm{vol}\\Omega)^\\frac{2}{n}}k^\\frac{2}{n}\\qquad\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1411.1135","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2014-11-05T03:06:46Z","cross_cats_sorted":["math.AP","math.SP"],"title_canon_sha256":"df35a48c918b134b27acc51d54ecfc2fd6986f16016d9cc66fa2b1924eac6cb7","abstract_canon_sha256":"9173f6718df50e0edcba27491a0080795345e912e06fa40db8ee57730b621589"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:08.767093Z","signature_b64":"DC5yNpgEaLV4i93oSD2fZhV0GWdhfrK0kRcaqIEfpY1CV0WXrP4i1fBxoR/KuSDYW+U7ORTWedvNutjMRPD/CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d9f18c97acaf8ae4a4bec8d32df8c15cf532ef404b6110990ca85ab0a9b59a6","last_reissued_at":"2026-05-18T02:38:08.766456Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:08.766456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of the P\\'{o}lya conjecture","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.AP","math.SP"],"primary_cat":"math.DG","authors_text":"Yue He","submitted_at":"2014-11-05T03:06:46Z","abstract_excerpt":"In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $\\Omega$ in $\\mathbb{R}^n$. It is well known that the $k$-th Dirichlet eigenvalue $\\lambda_k$ obeys the Weyl asymptotic formula, that is,\n\\[\n\\lambda_k\\sim\\frac{4\\pi^2}{(\\omega_n\\mathrm{vol}\\Omega)^\\frac{2}{n}}k^\\frac{2}{n}\\qquad\\hbox{as}\\quad k\\rightarrow\\infty,\n\\]\nwhere $\\mathrm{vol}\\Omega$ is the volume of $\\Omega$. In view of the above formula, P\\'{o}lya conjectured that\n\\[\n\\lambda_k\\gs\\frac{4\\pi^2}{(\\omega_n\\mathrm{vol}\\Omega)^\\frac{2}{n}}k^\\frac{2}{n}\\qquad\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1135","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1411.1135","created_at":"2026-05-18T02:38:08.766564+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.1135v2","created_at":"2026-05-18T02:38:08.766564+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.1135","created_at":"2026-05-18T02:38:08.766564+00:00"},{"alias_kind":"pith_short_12","alias_value":"JWPRRSL2ZL4K","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"JWPRRSL2ZL4K4SSL","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"JWPRRSL2","created_at":"2026-05-18T12:28:35.611951+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX","json":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX.json","graph_json":"https://pith.science/api/pith-number/JWPRRSL2ZL4K4SSL5SGTFX4MCX/graph.json","events_json":"https://pith.science/api/pith-number/JWPRRSL2ZL4K4SSL5SGTFX4MCX/events.json","paper":"https://pith.science/paper/JWPRRSL2"},"agent_actions":{"view_html":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX","download_json":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX.json","view_paper":"https://pith.science/paper/JWPRRSL2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.1135&json=true","fetch_graph":"https://pith.science/api/pith-number/JWPRRSL2ZL4K4SSL5SGTFX4MCX/graph.json","fetch_events":"https://pith.science/api/pith-number/JWPRRSL2ZL4K4SSL5SGTFX4MCX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/action/storage_attestation","attest_author":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/action/author_attestation","sign_citation":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/action/citation_signature","submit_replication":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/action/replication_record"}},"created_at":"2026-05-18T02:38:08.766564+00:00","updated_at":"2026-05-18T02:38:08.766564+00:00"}