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When $\\theta <0$, we use Harper's randomisation argument to introduce explicit conditioning to recover moment upper bounds consistent with critical normalisation predicted there. As an application, we prove that for $q(1-o(1))$ Dirichlet characters modulo $q$, $\\max_{|h| \\leq 1/2}|L(1/2+ih,\\chi)|\\ll \\frac{\\log(q)}{(\\log\\log(q))^{3/4+o(1)}}$, establishin"},"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":"2608.09906","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-08-10T17:52:05Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"9a48a6bc8fee181f084e9df5c28a6cd1905c6aa2fc02b7e044d721a17bf89ff1","abstract_canon_sha256":"9ae7daff174e4cdfb252e29d4a0890e9c91c6fe7edbce6dbfcb23603aa84819c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T02:25:25.000400Z","signature_b64":"SwZLijY0akpsVdiaW8vBFueMZRdBu0staUp3aLpDFviR3EsymnwHzg560QcJ0QKCfb7qtuEHryerswfgUpxGBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"208fba05fd1ba57f6a2095477ad18d43819e827a1475c642826e601c8f73beb1","last_reissued_at":"2026-08-11T02:25:24.998797Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T02:25:24.998797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the $\\beta=2$ Partition function for Dirichlet $L$-functions in the $q$-aspect","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Christopher Atherfold","submitted_at":"2026-08-10T17:52:05Z","abstract_excerpt":"We study the $\\beta=2$ partition function $\\int_{|h| \\leq \\log^{\\theta}(q)/2}|L(1/2+ih,\\chi)|^2dh$ for typical Dirichlet characters modulo a large prime $q$ and $\\theta \\in (-1/2,0]$ motivated by a $q$-analogue of the Saksman--Webb conjectures. 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As an application, we prove that for $q(1-o(1))$ Dirichlet characters modulo $q$, $\\max_{|h| \\leq 1/2}|L(1/2+ih,\\chi)|\\ll \\frac{\\log(q)}{(\\log\\log(q))^{3/4+o(1)}}$, establishin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.09906","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.09906/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":"2608.09906","created_at":"2026-08-11T02:25:24.999467+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.09906v1","created_at":"2026-08-11T02:25:24.999467+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.09906","created_at":"2026-08-11T02:25:24.999467+00:00"},{"alias_kind":"pith_short_12","alias_value":"ECH3UBP5DOSX","created_at":"2026-08-11T02:25:24.999467+00:00"},{"alias_kind":"pith_short_16","alias_value":"ECH3UBP5DOSX62RA","created_at":"2026-08-11T02:25:24.999467+00:00"},{"alias_kind":"pith_short_8","alias_value":"ECH3UBP5","created_at":"2026-08-11T02:25:24.999467+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/ECH3UBP5DOSX62RASVDXVUMNIO","json":"https://pith.science/pith/ECH3UBP5DOSX62RASVDXVUMNIO.json","graph_json":"https://pith.science/api/pith-number/ECH3UBP5DOSX62RASVDXVUMNIO/graph.json","events_json":"https://pith.science/api/pith-number/ECH3UBP5DOSX62RASVDXVUMNIO/events.json","paper":"https://pith.science/paper/ECH3UBP5"},"agent_actions":{"view_html":"https://pith.science/pith/ECH3UBP5DOSX62RASVDXVUMNIO","download_json":"https://pith.science/pith/ECH3UBP5DOSX62RASVDXVUMNIO.json","view_paper":"https://pith.science/paper/ECH3UBP5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.09906&json=true","fetch_graph":"https://pith.science/api/pith-number/ECH3UBP5DOSX62RASVDXVUMNIO/graph.json","fetch_events":"https://pith.science/api/pith-number/ECH3UBP5DOSX62RASVDXVUMNIO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ECH3UBP5DOSX62RASVDXVUMNIO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ECH3UBP5DOSX62RASVDXVUMNIO/action/storage_attestation","attest_author":"https://pith.science/pith/ECH3UBP5DOSX62RASVDXVUMNIO/action/author_attestation","sign_citation":"https://pith.science/pith/ECH3UBP5DOSX62RASVDXVUMNIO/action/citation_signature","submit_replication":"https://pith.science/pith/ECH3UBP5DOSX62RASVDXVUMNIO/action/replication_record"}},"created_at":"2026-08-11T02:25:24.999467+00:00","updated_at":"2026-08-11T02:25:24.999467+00:00"}