{"paper":{"title":"A Simple Proof of the Mean Value of $\\left|K_{2}(\\mathcal{O})\\right|$ in Function Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Julio Andrade","submitted_at":"2015-04-23T14:41:06Z","abstract_excerpt":"Let $F$ be a finite field of odd cardinality $q$, $A=F[T]$ the polynomial ring over $F$, $k=F(T)$ the rational function field over $F$ and $\\mathcal{H}$ the set of square-free monic polynomials in $A$ of degree odd. If $D\\in\\mathcal{H}$, we denote by $\\mathcal{O}_{D}$ the integral closure of $A$ in $k(\\sqrt{D})$. In this note we give a simple proof for the average value of the size of the groups $K_{2}(\\mathcal{O}_{D})$ as $D$ varies over the ensemble $\\mathcal{H}$ and $q$ is kept fixed. The proof is based on character sums estimates and in the use of the Riemann hypothesis for curves over fin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06205","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"}