{"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"}