{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:TTK2ZQEVFNSBANSYPR3IMJP3AI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3cd9614120124a61849cd6c54ce80ef34be7a7ef0c9cada6209b84c638a04627","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-12T19:08:39Z","title_canon_sha256":"07c4bdcd50e2a09fed4571702abc544374f99c0a6e9bbb75501fb58257fd49d0"},"schema_version":"1.0","source":{"id":"1810.05684","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.05684","created_at":"2026-05-18T00:03:27Z"},{"alias_kind":"arxiv_version","alias_value":"1810.05684v1","created_at":"2026-05-18T00:03:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.05684","created_at":"2026-05-18T00:03:27Z"},{"alias_kind":"pith_short_12","alias_value":"TTK2ZQEVFNSB","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"TTK2ZQEVFNSBANSY","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"TTK2ZQEV","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:47c5e9db4f556f8d1ace8037067364af9440bb9b09108e02eb0f4c92e669156a","target":"graph","created_at":"2026-05-18T00:03:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\chi$ range over the $(p-1)/2$ even Dirichlet characters modulo a prime $p$ and denote by $\\theta (x,\\chi)$ the associated theta series. The asymptotic behaviour of the second and fourth moments proved by Louboutin and the author implies that there exists at least $ \\gg p/ \\log p$ characters such that the associated theta function does not vanish at a fixed point. Constructing a suitable mollifier, we improve this result and show that there exists at least $ \\gg p/ \\sqrt{\\log p}$ characters such that $\\theta(x,\\chi) \\neq 0$ for any $x>0$. We give similar results for odd Dirichlet characte","authors_text":"Marc Munsch","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-12T19:08:39Z","title":"Non vanishing of theta functions and sets of small multiplicative energy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.05684","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ce57e820a953ffbb55084efd615ad04134096bdfabbdfb85a1b9ad97f340c716","target":"record","created_at":"2026-05-18T00:03:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3cd9614120124a61849cd6c54ce80ef34be7a7ef0c9cada6209b84c638a04627","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-12T19:08:39Z","title_canon_sha256":"07c4bdcd50e2a09fed4571702abc544374f99c0a6e9bbb75501fb58257fd49d0"},"schema_version":"1.0","source":{"id":"1810.05684","kind":"arxiv","version":1}},"canonical_sha256":"9cd5acc0952b641036587c768625fb023b67ffb0f2c009b8175e2eac0c4a85a8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9cd5acc0952b641036587c768625fb023b67ffb0f2c009b8175e2eac0c4a85a8","first_computed_at":"2026-05-18T00:03:27.246307Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:03:27.246307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PoDrImO5S4fIphKfzkbfmDShzgCPvmFQlAEWcp1llETOjIYHhbSU1KhRZhXl29/moMUgbOxcwclChoplhBWKDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:03:27.246773Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.05684","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce57e820a953ffbb55084efd615ad04134096bdfabbdfb85a1b9ad97f340c716","sha256:47c5e9db4f556f8d1ace8037067364af9440bb9b09108e02eb0f4c92e669156a"],"state_sha256":"c8c1d19c779d9cca4f273bf3b36f96c454eda7c72ff741c1c0f289e8d6932097"}