{"paper":{"title":"Positivity and tails of Jacobi theta series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Nian Hong Zhou","submitted_at":"2026-07-13T00:26:41Z","abstract_excerpt":"Using elementary $q$-series manipulations, we establish a positivity property for the tails of the Jacobi theta series. Specifically, for integers $k\\ge 1$ and $n\\ge 0$, define \\[ \\sum_{n\\ge0}\\sum_{m\\in\\mathbb{Z}}J_{k,n}(m)z^m q^{n} =\n\\frac{(-1)^k q^{-\\binom{k+1}{2}}}{(z)_{\\infty}(q/z)_\\infty} \\sum_{j\\ge k}(-1)^jq^{\\binom{j+1}{2}}z^{-j}(1-z^{2j+1}), \\] where $(a)_\\infty:=\\prod_{n\\ge0}(1-aq^n)$ denotes the $q$-shifted factorial. We prove that for all integers $k\\ge 1$ and $n\\ge 0$, the coefficients $J_{k,n}(m)$ are positive for all integers $-(k+n)\\le m\\le k+n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10968","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/2607.10968/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"}