{"paper":{"title":"On Witten's extremal partition functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ken Ono, Larry Rolen","submitted_at":"2018-07-02T02:56:48Z","abstract_excerpt":"In his famous 2007 paper on three dimensional quantum gravity, Witten\n  defined candidates for the partition functions $$Z_k(q)=\\sum_{n=-k}^{\\infty}w_k(n)q^n$$ of potential extremal CFTs with central charges of the form $c=24k$. Although such CFTs remain elusive, he proved that these modular functions are well-defined. In this note, we point out several explicit representations of these functions. These involve the partition function $p(n)$, Faber polynomials, traces of singular moduli, and Rademacher sums. Furthermore, for each prime $p\\leq 11$, the $p$ series $Z_k(q)$, where $k\\in \\{1, \\dots"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.00444","kind":"arxiv","version":3},"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"}