{"paper":{"title":"An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Brandon Alberts","submitted_at":"2025-08-28T14:13:44Z","abstract_excerpt":"Given a Dirichlet series $L(s) = \\sum a_n n^{-s}$, the asymptotic growth rate of $\\sum_{n\\le X} a_n$ can be determined by a Tauberian theorem. Bounds on the error term are typically controlled by the size of $|L(\\sigma+it)|$ for fixed real part $\\sigma$. We modify this approach to prove new Tauberian theorems with error terms depending only on the average size of $L(\\sigma+it)$ as $t$ varies, and we take care to track explicit dependence on various parameters. This often leads to stronger error bounds, and introduces strong connections between asymptotic counting problems and moments of $L$-fu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.20814","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.20814/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"}