{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:GNQWSVS3BMUZHH2RM2ZKOQSAGE","short_pith_number":"pith:GNQWSVS3","schema_version":"1.0","canonical_sha256":"336169565b0b29939f5166b2a7424031207fbc7b1e137523540b0007138eadd4","source":{"kind":"arxiv","id":"2608.07399","version":1},"attestation_state":"computed","paper":{"title":"Siegel zeros and small gaps between zeros of the Riemann zeta function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andriy Bondarenko, Winston Heap","submitted_at":"2026-08-07T16:42:11Z","abstract_excerpt":"On assuming the Riemann Hypothesis, we show that Siegel zeros imply the existence of gaps between the zeros of the Riemann zeta function less than 1/2 the normalised length. Specifically, we show that an infinite family of Siegel zeros implies $\\liminf_{n\\to\\infty}(\\gamma_{n+1}-\\gamma_n)\\log(\\gamma_n)/2\\pi< 0.4733$ on RH. This refutes the existence of certain strong alternative hypotheses under these assumptions. Our arguments incorporate long Dirichlet polynomials of length $T^{17/14-\\varepsilon}$ into the Montgomery--Odlyzko method."},"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":"2608.07399","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-08-07T16:42:11Z","cross_cats_sorted":[],"title_canon_sha256":"d735f149b947e13e541195cc3dacd580c6d5425e778a9023349844569be84237","abstract_canon_sha256":"7608432ab0e393271d97e241a6cf8e4950c7b12721de4c30c05cfc1a6f5effc6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-10T01:15:15.855835Z","signature_b64":"7dayIC1wIqTl4L/nM8X3FPbtG5UHR7j5XE+C2YCymiVCMLrOat4Dq+oqiULrn5EeLRS3FouqN5Wcxc0/xfFaDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"336169565b0b29939f5166b2a7424031207fbc7b1e137523540b0007138eadd4","last_reissued_at":"2026-08-10T01:15:15.853413Z","signature_status":"signed_v1","first_computed_at":"2026-08-10T01:15:15.853413Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Siegel zeros and small gaps between zeros of the Riemann zeta function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andriy Bondarenko, Winston Heap","submitted_at":"2026-08-07T16:42:11Z","abstract_excerpt":"On assuming the Riemann Hypothesis, we show that Siegel zeros imply the existence of gaps between the zeros of the Riemann zeta function less than 1/2 the normalised length. Specifically, we show that an infinite family of Siegel zeros implies $\\liminf_{n\\to\\infty}(\\gamma_{n+1}-\\gamma_n)\\log(\\gamma_n)/2\\pi< 0.4733$ on RH. This refutes the existence of certain strong alternative hypotheses under these assumptions. Our arguments incorporate long Dirichlet polynomials of length $T^{17/14-\\varepsilon}$ into the Montgomery--Odlyzko method."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.07399","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/2608.07399/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":"2608.07399","created_at":"2026-08-10T01:15:15.854415+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.07399v1","created_at":"2026-08-10T01:15:15.854415+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.07399","created_at":"2026-08-10T01:15:15.854415+00:00"},{"alias_kind":"pith_short_12","alias_value":"GNQWSVS3BMUZ","created_at":"2026-08-10T01:15:15.854415+00:00"},{"alias_kind":"pith_short_16","alias_value":"GNQWSVS3BMUZHH2R","created_at":"2026-08-10T01:15:15.854415+00:00"},{"alias_kind":"pith_short_8","alias_value":"GNQWSVS3","created_at":"2026-08-10T01:15:15.854415+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/GNQWSVS3BMUZHH2RM2ZKOQSAGE","json":"https://pith.science/pith/GNQWSVS3BMUZHH2RM2ZKOQSAGE.json","graph_json":"https://pith.science/api/pith-number/GNQWSVS3BMUZHH2RM2ZKOQSAGE/graph.json","events_json":"https://pith.science/api/pith-number/GNQWSVS3BMUZHH2RM2ZKOQSAGE/events.json","paper":"https://pith.science/paper/GNQWSVS3"},"agent_actions":{"view_html":"https://pith.science/pith/GNQWSVS3BMUZHH2RM2ZKOQSAGE","download_json":"https://pith.science/pith/GNQWSVS3BMUZHH2RM2ZKOQSAGE.json","view_paper":"https://pith.science/paper/GNQWSVS3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.07399&json=true","fetch_graph":"https://pith.science/api/pith-number/GNQWSVS3BMUZHH2RM2ZKOQSAGE/graph.json","fetch_events":"https://pith.science/api/pith-number/GNQWSVS3BMUZHH2RM2ZKOQSAGE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GNQWSVS3BMUZHH2RM2ZKOQSAGE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GNQWSVS3BMUZHH2RM2ZKOQSAGE/action/storage_attestation","attest_author":"https://pith.science/pith/GNQWSVS3BMUZHH2RM2ZKOQSAGE/action/author_attestation","sign_citation":"https://pith.science/pith/GNQWSVS3BMUZHH2RM2ZKOQSAGE/action/citation_signature","submit_replication":"https://pith.science/pith/GNQWSVS3BMUZHH2RM2ZKOQSAGE/action/replication_record"}},"created_at":"2026-08-10T01:15:15.854415+00:00","updated_at":"2026-08-10T01:15:15.854415+00:00"}