{"paper":{"title":"Average Error for Spectral Asymptotics on Surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Robert S. Strichartz","submitted_at":"2013-01-21T19:07:46Z","abstract_excerpt":"Let $N(t)$ denote the eigenvalue counting funtion of the Laplacian on a compact surface of constant nonnegative curvature, with or without boundary. We define a refined asymptotic formula $\\tilde{N}(t)=At+Bt^{1/2}+C$, where the constants are expressed in terms of the geometry of the surface and its boundary, and consider the average error $A(t) = \\frac{1}{t} \\int_0^{t}D(s)ds$ for $D(t) = N(t) - \\tilde{N}(t)$. We present a conjecture for the asymptotic behavior of $A(t)$, and study some examples that support the conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.4963","kind":"arxiv","version":1},"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"}