{"paper":{"title":"P\\'olya's conjecture up to $\\epsilon$-loss and quantitative estimates for the remainder of Weyl's law","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.AP","math.CA","math.MP"],"primary_cat":"math.SP","authors_text":"Fanghua Lin, Renjin Jiang","submitted_at":"2025-07-06T09:18:20Z","abstract_excerpt":"Let $\\Omega\\subset\\mathbb{R}^n$ be a bounded Lipschitz domain. For any $\\epsilon\\in (0,1)$ we show that for any Dirichlet eigenvalue $\\lambda_k(\\Omega)>\\Lambda(\\epsilon,\\Omega)$, it holds \\begin{align*} k&\\le (1+\\epsilon)\\frac{|\\Omega|\\omega(n)}{(2\\pi)^n}\\lambda_k(\\Omega)^{n/2}, \\end{align*} where $\\Lambda(\\epsilon,\\Omega)$ is given explicitly. This reduces the $\\epsilon$-loss version of P\\'olya's conjecture to a computational problem. This estimate is based on quantitative estimates on the remainder of the Weyl law with explicit constants, which we give a new proof without using Neumann eigen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.04307","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.04307/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}