{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:2E7SD6N5EFUFSA3MZIBDWYBHLM","short_pith_number":"pith:2E7SD6N5","schema_version":"1.0","canonical_sha256":"d13f21f9bd216859036cca023b60275b39156e3d305b70838c400f0e7dc0701c","source":{"kind":"arxiv","id":"1612.01698","version":1},"attestation_state":"computed","paper":{"title":"Moments of Hardy's function over short intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aleksandar Ivi\\'c","submitted_at":"2016-12-06T08:18:34Z","abstract_excerpt":"Let as usual $Z(t) = \\zeta(1/2+it)\\chi^{-1/2}(1/2+it)$ denote Hardy's function, where $\\zeta(s) = \\chi(s)\\zeta(1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z'(t)$ are proved. It is also proved that $$ H(\\log T)^{k^2} \\ll_{k,\\alpha} \\sum_{T<\\gamma\\le T+H}\\max_{\\gamma\\le \\tau_\\gamma\\le \\gamma^+} |\\zeta(1/2 + i\\tau_\\gamma)|^{2k} \\ll_{k,\\alpha} H(\\log T)^{k^2}. $$ Here $k>1$ is a fixed integer, $\\gamma, \\gamma^+$ denote ordinates of consecutive complex zeros of $\\zeta(s)$ and $T^\\alpha \\le H \\le T$, where $\\alpha$ is a fixed constant such t"},"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":"1612.01698","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-06T08:18:34Z","cross_cats_sorted":[],"title_canon_sha256":"cbd6282149008b545cc01bca7ec849610a591d166732fa2f41bcb429d4b93736","abstract_canon_sha256":"5e086091bf98815044a1d014f2a29c12af41372494082335d049591c79d5e573"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:47.808081Z","signature_b64":"pvyRc6SLkgeworFShoc+4ZcBTFpSjWNYV7/JfAridJBbZMaz2jY5MVg6z8kIR9E4h4YOisEOlNNX8wHF9QUqAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d13f21f9bd216859036cca023b60275b39156e3d305b70838c400f0e7dc0701c","last_reissued_at":"2026-05-18T00:55:47.807607Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:47.807607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Moments of Hardy's function over short intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aleksandar Ivi\\'c","submitted_at":"2016-12-06T08:18:34Z","abstract_excerpt":"Let as usual $Z(t) = \\zeta(1/2+it)\\chi^{-1/2}(1/2+it)$ denote Hardy's function, where $\\zeta(s) = \\chi(s)\\zeta(1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z'(t)$ are proved. It is also proved that $$ H(\\log T)^{k^2} \\ll_{k,\\alpha} \\sum_{T<\\gamma\\le T+H}\\max_{\\gamma\\le \\tau_\\gamma\\le \\gamma^+} |\\zeta(1/2 + i\\tau_\\gamma)|^{2k} \\ll_{k,\\alpha} H(\\log T)^{k^2}. $$ Here $k>1$ is a fixed integer, $\\gamma, \\gamma^+$ denote ordinates of consecutive complex zeros of $\\zeta(s)$ and $T^\\alpha \\le H \\le T$, where $\\alpha$ is a fixed constant such t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01698","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":""},"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":"1612.01698","created_at":"2026-05-18T00:55:47.807692+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.01698v1","created_at":"2026-05-18T00:55:47.807692+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.01698","created_at":"2026-05-18T00:55:47.807692+00:00"},{"alias_kind":"pith_short_12","alias_value":"2E7SD6N5EFUF","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"2E7SD6N5EFUFSA3M","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"2E7SD6N5","created_at":"2026-05-18T12:29:55.572404+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/2E7SD6N5EFUFSA3MZIBDWYBHLM","json":"https://pith.science/pith/2E7SD6N5EFUFSA3MZIBDWYBHLM.json","graph_json":"https://pith.science/api/pith-number/2E7SD6N5EFUFSA3MZIBDWYBHLM/graph.json","events_json":"https://pith.science/api/pith-number/2E7SD6N5EFUFSA3MZIBDWYBHLM/events.json","paper":"https://pith.science/paper/2E7SD6N5"},"agent_actions":{"view_html":"https://pith.science/pith/2E7SD6N5EFUFSA3MZIBDWYBHLM","download_json":"https://pith.science/pith/2E7SD6N5EFUFSA3MZIBDWYBHLM.json","view_paper":"https://pith.science/paper/2E7SD6N5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.01698&json=true","fetch_graph":"https://pith.science/api/pith-number/2E7SD6N5EFUFSA3MZIBDWYBHLM/graph.json","fetch_events":"https://pith.science/api/pith-number/2E7SD6N5EFUFSA3MZIBDWYBHLM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2E7SD6N5EFUFSA3MZIBDWYBHLM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2E7SD6N5EFUFSA3MZIBDWYBHLM/action/storage_attestation","attest_author":"https://pith.science/pith/2E7SD6N5EFUFSA3MZIBDWYBHLM/action/author_attestation","sign_citation":"https://pith.science/pith/2E7SD6N5EFUFSA3MZIBDWYBHLM/action/citation_signature","submit_replication":"https://pith.science/pith/2E7SD6N5EFUFSA3MZIBDWYBHLM/action/replication_record"}},"created_at":"2026-05-18T00:55:47.807692+00:00","updated_at":"2026-05-18T00:55:47.807692+00:00"}