{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:ONSK3KDKJI3IJO4SQWX376RN7L","short_pith_number":"pith:ONSK3KDK","schema_version":"1.0","canonical_sha256":"7364ada86a4a3684bb9285afbffa2dfaed6375da021819f02b8a889ddcca96cb","source":{"kind":"arxiv","id":"2411.13255","version":2},"attestation_state":"computed","paper":{"title":"Note on the $a$-points of the Riemann zeta function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Min-Jie Luo, Peng-Cheng Hang","submitted_at":"2024-11-20T12:14:02Z","abstract_excerpt":"For any $a\\in\\mathbb{C}$, the zeros of $\\zeta(s)-a$, denoted by $\\rho_a=\\beta_a+i\\gamma_a$, are called $a$-points of the Riemann zeta function $\\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\\zeta(s)$ shown by Garunk\\v{s}tis and Steuding. We then deduce an asymptotic of the sum \\[S_T(a,\\delta)=\\sum_{\\tau<\\gamma_a\\leqslant T}\\zeta'(\\rho_a+i\\delta)X^{\\rho_a},\\quad T\\to\\infty,\\] where $0\\ne\\delta=\\frac{2\\pi\\alpha}{\\log\\frac{T}{2\\pi X}}\\ll 1$, and $X>0$ and $\\tau\\geqslant|\\delta|+1$ are fixed.\n  We also find the interesting varied behavior of $S_T(a,\\delta)$ i"},"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":"2411.13255","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-20T12:14:02Z","cross_cats_sorted":[],"title_canon_sha256":"9467695e3b44d548b365428454ff015b874c225a6a7e0c22bd840630652ec9cd","abstract_canon_sha256":"f726231a6b787f31aab9ed8b6aebfe7fe1d27fa3191852cc2aca466727d48dbf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:38:33.271486Z","signature_b64":"ye2Gtg6oulVpKvzHeueBIB8Ly/54h67MITeZ1ijYc3W6F1jv91PZ/3zWTqFNrNlW8x0qaCovxFR8NGWtQ77dBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7364ada86a4a3684bb9285afbffa2dfaed6375da021819f02b8a889ddcca96cb","last_reissued_at":"2026-07-05T09:38:33.271001Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:38:33.271001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Note on the $a$-points of the Riemann zeta function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Min-Jie Luo, Peng-Cheng Hang","submitted_at":"2024-11-20T12:14:02Z","abstract_excerpt":"For any $a\\in\\mathbb{C}$, the zeros of $\\zeta(s)-a$, denoted by $\\rho_a=\\beta_a+i\\gamma_a$, are called $a$-points of the Riemann zeta function $\\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\\zeta(s)$ shown by Garunk\\v{s}tis and Steuding. We then deduce an asymptotic of the sum \\[S_T(a,\\delta)=\\sum_{\\tau<\\gamma_a\\leqslant T}\\zeta'(\\rho_a+i\\delta)X^{\\rho_a},\\quad T\\to\\infty,\\] where $0\\ne\\delta=\\frac{2\\pi\\alpha}{\\log\\frac{T}{2\\pi X}}\\ll 1$, and $X>0$ and $\\tau\\geqslant|\\delta|+1$ are fixed.\n  We also find the interesting varied behavior of $S_T(a,\\delta)$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.13255","kind":"arxiv","version":2},"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/2411.13255/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":"2411.13255","created_at":"2026-07-05T09:38:33.271061+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.13255v2","created_at":"2026-07-05T09:38:33.271061+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.13255","created_at":"2026-07-05T09:38:33.271061+00:00"},{"alias_kind":"pith_short_12","alias_value":"ONSK3KDKJI3I","created_at":"2026-07-05T09:38:33.271061+00:00"},{"alias_kind":"pith_short_16","alias_value":"ONSK3KDKJI3IJO4S","created_at":"2026-07-05T09:38:33.271061+00:00"},{"alias_kind":"pith_short_8","alias_value":"ONSK3KDK","created_at":"2026-07-05T09:38:33.271061+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/ONSK3KDKJI3IJO4SQWX376RN7L","json":"https://pith.science/pith/ONSK3KDKJI3IJO4SQWX376RN7L.json","graph_json":"https://pith.science/api/pith-number/ONSK3KDKJI3IJO4SQWX376RN7L/graph.json","events_json":"https://pith.science/api/pith-number/ONSK3KDKJI3IJO4SQWX376RN7L/events.json","paper":"https://pith.science/paper/ONSK3KDK"},"agent_actions":{"view_html":"https://pith.science/pith/ONSK3KDKJI3IJO4SQWX376RN7L","download_json":"https://pith.science/pith/ONSK3KDKJI3IJO4SQWX376RN7L.json","view_paper":"https://pith.science/paper/ONSK3KDK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.13255&json=true","fetch_graph":"https://pith.science/api/pith-number/ONSK3KDKJI3IJO4SQWX376RN7L/graph.json","fetch_events":"https://pith.science/api/pith-number/ONSK3KDKJI3IJO4SQWX376RN7L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ONSK3KDKJI3IJO4SQWX376RN7L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ONSK3KDKJI3IJO4SQWX376RN7L/action/storage_attestation","attest_author":"https://pith.science/pith/ONSK3KDKJI3IJO4SQWX376RN7L/action/author_attestation","sign_citation":"https://pith.science/pith/ONSK3KDKJI3IJO4SQWX376RN7L/action/citation_signature","submit_replication":"https://pith.science/pith/ONSK3KDKJI3IJO4SQWX376RN7L/action/replication_record"}},"created_at":"2026-07-05T09:38:33.271061+00:00","updated_at":"2026-07-05T09:38:33.271061+00:00"}