{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:KZL75AKNRLEDNSAJNLGNT56JBT","short_pith_number":"pith:KZL75AKN","schema_version":"1.0","canonical_sha256":"5657fe814d8ac836c8096accd9f7c90cec9ee7e9d410e3eb54cf23e48d5e83a9","source":{"kind":"arxiv","id":"2303.10123","version":2},"attestation_state":"computed","paper":{"title":"Correlations of the Riemann zeta function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael J. Curran","submitted_at":"2023-03-17T17:03:50Z","abstract_excerpt":"Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function \\[ M_{\\alpha,{\\beta}}(T) = \\int_T^{2T} \\prod_{k = 1}^m |\\zeta(\\tfrac{1}{2} + i (t + \\alpha_k))|^{2 \\beta_k} dt \\] introduced by Chandee, where ${\\alpha} = {\\alpha}(T) = (\\alpha_1, \\ldots, \\alpha_m)$ and ${\\beta} = (\\beta_1 \\ldots , \\beta_m)$ satisfy $|\\alpha_k| \\leq T/2$ and $\\beta_k\\geq 0$. We shall prove that \\[ M_{{\\alpha},{\\beta}}(T) \\ll_{{\\beta}} T (\\log T)^{\\beta_1^2 + \\cdots + \\beta_m^2} \\prod_{1\\leq j < k \\leq m} |\\zeta(1 + i(\\alpha_j - \\alpha_k) + 1/ \\log T )|^{2\\beta_j \\beta_k}. \\] This improves "},"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":"2303.10123","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2023-03-17T17:03:50Z","cross_cats_sorted":[],"title_canon_sha256":"f69d60f758b5dae34544d10b281cf4fa73dd80e9bf4c4f999c773aadccfcf6e4","abstract_canon_sha256":"4f477e19bbfa97502e1ab406ff12628b721ffbc3119aa554493bec9d031042ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:19:18.257102Z","signature_b64":"8DmEpvEKbsI5lhvdhmW78QFRF7BliTZ7G6Lkdo2WyjyeamP8Ue47Cm+OzRw+qdDEeio+2mdIFrgb027lnXnJDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5657fe814d8ac836c8096accd9f7c90cec9ee7e9d410e3eb54cf23e48d5e83a9","last_reissued_at":"2026-07-05T08:19:18.256590Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:19:18.256590Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Correlations of the Riemann zeta function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael J. Curran","submitted_at":"2023-03-17T17:03:50Z","abstract_excerpt":"Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function \\[ M_{\\alpha,{\\beta}}(T) = \\int_T^{2T} \\prod_{k = 1}^m |\\zeta(\\tfrac{1}{2} + i (t + \\alpha_k))|^{2 \\beta_k} dt \\] introduced by Chandee, where ${\\alpha} = {\\alpha}(T) = (\\alpha_1, \\ldots, \\alpha_m)$ and ${\\beta} = (\\beta_1 \\ldots , \\beta_m)$ satisfy $|\\alpha_k| \\leq T/2$ and $\\beta_k\\geq 0$. We shall prove that \\[ M_{{\\alpha},{\\beta}}(T) \\ll_{{\\beta}} T (\\log T)^{\\beta_1^2 + \\cdots + \\beta_m^2} \\prod_{1\\leq j < k \\leq m} |\\zeta(1 + i(\\alpha_j - \\alpha_k) + 1/ \\log T )|^{2\\beta_j \\beta_k}. \\] This improves "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.10123","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/2303.10123/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":"2303.10123","created_at":"2026-07-05T08:19:18.256653+00:00"},{"alias_kind":"arxiv_version","alias_value":"2303.10123v2","created_at":"2026-07-05T08:19:18.256653+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.10123","created_at":"2026-07-05T08:19:18.256653+00:00"},{"alias_kind":"pith_short_12","alias_value":"KZL75AKNRLED","created_at":"2026-07-05T08:19:18.256653+00:00"},{"alias_kind":"pith_short_16","alias_value":"KZL75AKNRLEDNSAJ","created_at":"2026-07-05T08:19:18.256653+00:00"},{"alias_kind":"pith_short_8","alias_value":"KZL75AKN","created_at":"2026-07-05T08:19:18.256653+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.09334","citing_title":"Lower bounds for high moments of zeta sums","ref_index":2,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KZL75AKNRLEDNSAJNLGNT56JBT","json":"https://pith.science/pith/KZL75AKNRLEDNSAJNLGNT56JBT.json","graph_json":"https://pith.science/api/pith-number/KZL75AKNRLEDNSAJNLGNT56JBT/graph.json","events_json":"https://pith.science/api/pith-number/KZL75AKNRLEDNSAJNLGNT56JBT/events.json","paper":"https://pith.science/paper/KZL75AKN"},"agent_actions":{"view_html":"https://pith.science/pith/KZL75AKNRLEDNSAJNLGNT56JBT","download_json":"https://pith.science/pith/KZL75AKNRLEDNSAJNLGNT56JBT.json","view_paper":"https://pith.science/paper/KZL75AKN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2303.10123&json=true","fetch_graph":"https://pith.science/api/pith-number/KZL75AKNRLEDNSAJNLGNT56JBT/graph.json","fetch_events":"https://pith.science/api/pith-number/KZL75AKNRLEDNSAJNLGNT56JBT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KZL75AKNRLEDNSAJNLGNT56JBT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KZL75AKNRLEDNSAJNLGNT56JBT/action/storage_attestation","attest_author":"https://pith.science/pith/KZL75AKNRLEDNSAJNLGNT56JBT/action/author_attestation","sign_citation":"https://pith.science/pith/KZL75AKNRLEDNSAJNLGNT56JBT/action/citation_signature","submit_replication":"https://pith.science/pith/KZL75AKNRLEDNSAJNLGNT56JBT/action/replication_record"}},"created_at":"2026-07-05T08:19:18.256653+00:00","updated_at":"2026-07-05T08:19:18.256653+00:00"}