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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 $-(k+n)\\le m\\le k+n$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.10968","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-13T00:26:41Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e4fe5a28acbcc3ebc6865a2101def30f990e690a98f4957fd4469c3f75e52577","abstract_canon_sha256":"17cc80df62166ad050b85fb555405e4866e541c96345f1f432698db268c80f8d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:21:49.990645Z","signature_b64":"S01DJh6k9oMbL/JoBwbcx3j/rF05hCzqH8lGRXZB1yqgf0gnAIojDwz0YKdUwKgz5iRNX3s+odkh4NEDsqe5BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"361e9487ccc0f4363fcaaa7b3a352f2f64a28e9a9c9a7dd438356866fa698b0d","last_reissued_at":"2026-07-14T01:21:49.989816Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:21:49.989816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.10968","created_at":"2026-07-14T01:21:49.990244+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.10968v1","created_at":"2026-07-14T01:21:49.990244+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10968","created_at":"2026-07-14T01:21:49.990244+00:00"},{"alias_kind":"pith_short_12","alias_value":"GYPJJB6MYD2D","created_at":"2026-07-14T01:21:49.990244+00:00"},{"alias_kind":"pith_short_16","alias_value":"GYPJJB6MYD2DMP6K","created_at":"2026-07-14T01:21:49.990244+00:00"},{"alias_kind":"pith_short_8","alias_value":"GYPJJB6M","created_at":"2026-07-14T01:21:49.990244+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GYPJJB6MYD2DMP6KVJ5TUNJPF5","json":"https://pith.science/pith/GYPJJB6MYD2DMP6KVJ5TUNJPF5.json","graph_json":"https://pith.science/api/pith-number/GYPJJB6MYD2DMP6KVJ5TUNJPF5/graph.json","events_json":"https://pith.science/api/pith-number/GYPJJB6MYD2DMP6KVJ5TUNJPF5/events.json","paper":"https://pith.science/paper/GYPJJB6M"},"agent_actions":{"view_html":"https://pith.science/pith/GYPJJB6MYD2DMP6KVJ5TUNJPF5","download_json":"https://pith.science/pith/GYPJJB6MYD2DMP6KVJ5TUNJPF5.json","view_paper":"https://pith.science/paper/GYPJJB6M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.10968&json=true","fetch_graph":"https://pith.science/api/pith-number/GYPJJB6MYD2DMP6KVJ5TUNJPF5/graph.json","fetch_events":"https://pith.science/api/pith-number/GYPJJB6MYD2DMP6KVJ5TUNJPF5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GYPJJB6MYD2DMP6KVJ5TUNJPF5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GYPJJB6MYD2DMP6KVJ5TUNJPF5/action/storage_attestation","attest_author":"https://pith.science/pith/GYPJJB6MYD2DMP6KVJ5TUNJPF5/action/author_attestation","sign_citation":"https://pith.science/pith/GYPJJB6MYD2DMP6KVJ5TUNJPF5/action/citation_signature","submit_replication":"https://pith.science/pith/GYPJJB6MYD2DMP6KVJ5TUNJPF5/action/replication_record"}},"created_at":"2026-07-14T01:21:49.990244+00:00","updated_at":"2026-07-14T01:21:49.990244+00:00"}