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This is sharp up to the value of the implicit constant.\n  Our proof builds on well known work of Soundararajan, who showed, assuming the Riemann Hypothesis, that \\int_{T}^{2T} |\\zeta(1/2+it)|^{2k} dt \\ll_{k,\\epsilon} T log^{k^{2}+\\epsilon} T for any fixed k \\geq 0 and \\epsilon > 0. Whereas Soundararajan bounded \\log|\\zeta(1/2+it)| by a single Dirichlet polynomial, and investigated how often it attains large values, we bound \\log|\\zeta(1/2+it)| 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":"1305.4618","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-05-20T19:41:52Z","cross_cats_sorted":[],"title_canon_sha256":"a5d079637b30b33039e6b3a344692ee01d6d2ec9d6d7b6400dd6ce6b248b4a99","abstract_canon_sha256":"3a0729e25a01c7f421beb4f9f539cf8a988911f1233550b46770e50aa8669c5f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:16.562835Z","signature_b64":"rqOQQJpXyHaQUBLrmmzqmgy/IGSY4gnZUkyJJnT1NI64TbKPFqzPPsdrqvCyy4NxjiImzALtT1oNT6FnG207Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53d55f3c6b7b22f0c539b7d4b4e43ee54d07373a11cc4e41c455b28a2da83dfa","last_reissued_at":"2026-05-18T03:25:16.562122Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:16.562122Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp conditional bounds for moments of the Riemann zeta function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adam J. Harper","submitted_at":"2013-05-20T19:41:52Z","abstract_excerpt":"We prove, assuming the Riemann Hypothesis, that \\int_{T}^{2T} |\\zeta(1/2+it)|^{2k} dt \\ll_{k} T log^{k^{2}} T for any fixed k \\geq 0 and all large T. This is sharp up to the value of the implicit constant.\n  Our proof builds on well known work of Soundararajan, who showed, assuming the Riemann Hypothesis, that \\int_{T}^{2T} |\\zeta(1/2+it)|^{2k} dt \\ll_{k,\\epsilon} T log^{k^{2}+\\epsilon} T for any fixed k \\geq 0 and \\epsilon > 0. Whereas Soundararajan bounded \\log|\\zeta(1/2+it)| by a single Dirichlet polynomial, and investigated how often it attains large values, we bound \\log|\\zeta(1/2+it)| by"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4618","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":"1305.4618","created_at":"2026-05-18T03:25:16.562224+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.4618v1","created_at":"2026-05-18T03:25:16.562224+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4618","created_at":"2026-05-18T03:25:16.562224+00:00"},{"alias_kind":"pith_short_12","alias_value":"KPKV6PDLPMRP","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_16","alias_value":"KPKV6PDLPMRPBRJZ","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_8","alias_value":"KPKV6PDL","created_at":"2026-05-18T12:27:51.066281+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":3,"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":5,"is_internal_anchor":false},{"citing_arxiv_id":"2604.11941","citing_title":"Simultaneous non-vanishing of Dirichlet L-functions","ref_index":22,"is_internal_anchor":false},{"citing_arxiv_id":"2605.03665","citing_title":"Joint extreme values of $L$-functions on and off the critical line","ref_index":16,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KPKV6PDLPMRPBRJZW7KLJZB64V","json":"https://pith.science/pith/KPKV6PDLPMRPBRJZW7KLJZB64V.json","graph_json":"https://pith.science/api/pith-number/KPKV6PDLPMRPBRJZW7KLJZB64V/graph.json","events_json":"https://pith.science/api/pith-number/KPKV6PDLPMRPBRJZW7KLJZB64V/events.json","paper":"https://pith.science/paper/KPKV6PDL"},"agent_actions":{"view_html":"https://pith.science/pith/KPKV6PDLPMRPBRJZW7KLJZB64V","download_json":"https://pith.science/pith/KPKV6PDLPMRPBRJZW7KLJZB64V.json","view_paper":"https://pith.science/paper/KPKV6PDL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.4618&json=true","fetch_graph":"https://pith.science/api/pith-number/KPKV6PDLPMRPBRJZW7KLJZB64V/graph.json","fetch_events":"https://pith.science/api/pith-number/KPKV6PDLPMRPBRJZW7KLJZB64V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KPKV6PDLPMRPBRJZW7KLJZB64V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KPKV6PDLPMRPBRJZW7KLJZB64V/action/storage_attestation","attest_author":"https://pith.science/pith/KPKV6PDLPMRPBRJZW7KLJZB64V/action/author_attestation","sign_citation":"https://pith.science/pith/KPKV6PDLPMRPBRJZW7KLJZB64V/action/citation_signature","submit_replication":"https://pith.science/pith/KPKV6PDLPMRPBRJZW7KLJZB64V/action/replication_record"}},"created_at":"2026-05-18T03:25:16.562224+00:00","updated_at":"2026-05-18T03:25:16.562224+00:00"}