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In particular, we prove that for $(1+o(1))T$ values of $T \\leq t \\leq 2T$ we have $\\max_{|h| \\leq 1/2} \\log|\\zeta(1/2+it+ih)| \\leq \\log\\log T - (3/4 + o(1))\\log\\log\\log T$, matching for the first time with both the leading and second order terms predicted by a conjecture of Fyodorov, Hiary and Keating.\n  The proofs work by approximating the zeta function in mean square by"},"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":"1906.05783","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-13T16:17:07Z","cross_cats_sorted":[],"title_canon_sha256":"144644ad5ac662062b6a7d50c90f88bffe4984ceb536d5342f67c2a715784750","abstract_canon_sha256":"e08de2772e41259c820d3fc456913ff5d6db3d0ad5c990dbdff42f7f50cdf24e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:24.405507Z","signature_b64":"nD1p6TmBwHU9q8GqBexJ6QvnFtRlYB7d+bdNcDoOa6EBRgXUflDbstkOK+hqz+eVcswmFNzCD1DXpuynUczeAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb7c5d53b2624dd584413bca24091208f3d083664bbeab89de2587bd7f0eebba","last_reissued_at":"2026-05-17T23:43:24.405096Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:24.405096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adam J. Harper","submitted_at":"2019-06-13T16:17:07Z","abstract_excerpt":"We investigate the ``partition function'' integrals $\\int_{-1/2}^{1/2} |\\zeta(1/2 + it + ih)|^2 dh$ for the critical exponent 2, and the local maxima $\\max_{|h| \\leq 1/2} |\\zeta(1/2 + it + ih)|$, as $T \\leq t \\leq 2T$ varies. In particular, we prove that for $(1+o(1))T$ values of $T \\leq t \\leq 2T$ we have $\\max_{|h| \\leq 1/2} \\log|\\zeta(1/2+it+ih)| \\leq \\log\\log T - (3/4 + o(1))\\log\\log\\log T$, matching for the first time with both the leading and second order terms predicted by a conjecture of Fyodorov, Hiary and Keating.\n  The proofs work by approximating the zeta function in mean square by"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.05783","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":"1906.05783","created_at":"2026-05-17T23:43:24.405150+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.05783v1","created_at":"2026-05-17T23:43:24.405150+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.05783","created_at":"2026-05-17T23:43:24.405150+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZN6F2U5SMJG5","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZN6F2U5SMJG5LBCB","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZN6F2U5S","created_at":"2026-05-18T12:33:33.725879+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.25579","citing_title":"Conditional Upper Bounds for Large Deviations and Moments of the Riemann Zeta Function","ref_index":6,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZN6F2U5SMJG5LBCBHPFCICISBD","json":"https://pith.science/pith/ZN6F2U5SMJG5LBCBHPFCICISBD.json","graph_json":"https://pith.science/api/pith-number/ZN6F2U5SMJG5LBCBHPFCICISBD/graph.json","events_json":"https://pith.science/api/pith-number/ZN6F2U5SMJG5LBCBHPFCICISBD/events.json","paper":"https://pith.science/paper/ZN6F2U5S"},"agent_actions":{"view_html":"https://pith.science/pith/ZN6F2U5SMJG5LBCBHPFCICISBD","download_json":"https://pith.science/pith/ZN6F2U5SMJG5LBCBHPFCICISBD.json","view_paper":"https://pith.science/paper/ZN6F2U5S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.05783&json=true","fetch_graph":"https://pith.science/api/pith-number/ZN6F2U5SMJG5LBCBHPFCICISBD/graph.json","fetch_events":"https://pith.science/api/pith-number/ZN6F2U5SMJG5LBCBHPFCICISBD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZN6F2U5SMJG5LBCBHPFCICISBD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZN6F2U5SMJG5LBCBHPFCICISBD/action/storage_attestation","attest_author":"https://pith.science/pith/ZN6F2U5SMJG5LBCBHPFCICISBD/action/author_attestation","sign_citation":"https://pith.science/pith/ZN6F2U5SMJG5LBCBHPFCICISBD/action/citation_signature","submit_replication":"https://pith.science/pith/ZN6F2U5SMJG5LBCBHPFCICISBD/action/replication_record"}},"created_at":"2026-05-17T23:43:24.405150+00:00","updated_at":"2026-05-17T23:43:24.405150+00:00"}