{"paper":{"title":"Nonvanishing of $L$--functions associated to fixed order characters over function fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexandra Florea, Chantal David, Matilde Lalin","submitted_at":"2025-06-09T14:43:47Z","abstract_excerpt":"We show that a positive proportion of the values $L(1/2,\\chi_c)$ are non-zero, where $\\chi_c$ is the $\\ell^{\\text{th}}$ residue symbol for $\\ell \\geq 3$ over $\\mathbb{F}_q[t]$, when averaging over square-free polynomials $c$ in $\\mathbb{F}_q[t]$, as $q \\equiv 1(\\textrm{mod}\\,{2\\ell})$ is fixed and the degree of $c$ goes to infinity. In the case of $\\ell=3$, we show that at least $1/6$ of $L(1/2,\\chi_c)\\neq 0$, while for $\\ell>3$, the proportion depends on the order of the character. This improves a previous result of Ellenberg, Li, and Shusterman showing that there are infinitely many $\\chi$ o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07815","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/2506.07815/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"}