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These match the better than squareroot cancellation upper bounds obtained in previous work of the author. We prove the same sharp lower bounds for the moments $\\frac{1}{T} \\int_{0}^{T} |\\sum_{n \\leq x} n^{it}|^{2q} dt$ of zeta sums, and more generally for moments of character sums $\\sum_{n \\leq x} h(n) \\chi(n)$ with suitably bounded multiplicative twist $h(n)$.\n  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":"2607.01184","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-01T17:17:10Z","cross_cats_sorted":[],"title_canon_sha256":"96f7ac2c248a5f99a7776f8e850e7ac31e422e0e3799a3297d890587dd45ed93","abstract_canon_sha256":"0135f76965b37cb375add72c530d9963a15b3ad5154570d9e4e322889fb2f0aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T01:18:31.961856Z","signature_b64":"GyV35Ryj2sQP6K+Y1BJsLEVztBAsNHBGFJkqxaEuM7NEjpe9+v7tEAEYv4ekA5pHvAyF3SGKQ8gGQBPedUenDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"203b337595b2a99e1eab8220d7eb42d39a215a76ae89a3144b70d74fa277b2ef","last_reissued_at":"2026-07-02T01:18:31.961384Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T01:18:31.961384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lower bounds for low moments of character sums, I: Short sums with general multiplicative weights","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adam J. 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We prove the same sharp lower bounds for the moments $\\frac{1}{T} \\int_{0}^{T} |\\sum_{n \\leq x} n^{it}|^{2q} dt$ of zeta sums, and more generally for moments of character sums $\\sum_{n \\leq x} h(n) \\chi(n)$ with suitably bounded multiplicative twist $h(n)$.\n  T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01184","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/2607.01184/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":"2607.01184","created_at":"2026-07-02T01:18:31.961440+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.01184v1","created_at":"2026-07-02T01:18:31.961440+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01184","created_at":"2026-07-02T01:18:31.961440+00:00"},{"alias_kind":"pith_short_12","alias_value":"EA5TG5MVWKUZ","created_at":"2026-07-02T01:18:31.961440+00:00"},{"alias_kind":"pith_short_16","alias_value":"EA5TG5MVWKUZ4HVL","created_at":"2026-07-02T01:18:31.961440+00:00"},{"alias_kind":"pith_short_8","alias_value":"EA5TG5MV","created_at":"2026-07-02T01:18:31.961440+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/EA5TG5MVWKUZ4HVLQIQNP22C2O","json":"https://pith.science/pith/EA5TG5MVWKUZ4HVLQIQNP22C2O.json","graph_json":"https://pith.science/api/pith-number/EA5TG5MVWKUZ4HVLQIQNP22C2O/graph.json","events_json":"https://pith.science/api/pith-number/EA5TG5MVWKUZ4HVLQIQNP22C2O/events.json","paper":"https://pith.science/paper/EA5TG5MV"},"agent_actions":{"view_html":"https://pith.science/pith/EA5TG5MVWKUZ4HVLQIQNP22C2O","download_json":"https://pith.science/pith/EA5TG5MVWKUZ4HVLQIQNP22C2O.json","view_paper":"https://pith.science/paper/EA5TG5MV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.01184&json=true","fetch_graph":"https://pith.science/api/pith-number/EA5TG5MVWKUZ4HVLQIQNP22C2O/graph.json","fetch_events":"https://pith.science/api/pith-number/EA5TG5MVWKUZ4HVLQIQNP22C2O/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EA5TG5MVWKUZ4HVLQIQNP22C2O/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EA5TG5MVWKUZ4HVLQIQNP22C2O/action/storage_attestation","attest_author":"https://pith.science/pith/EA5TG5MVWKUZ4HVLQIQNP22C2O/action/author_attestation","sign_citation":"https://pith.science/pith/EA5TG5MVWKUZ4HVLQIQNP22C2O/action/citation_signature","submit_replication":"https://pith.science/pith/EA5TG5MVWKUZ4HVLQIQNP22C2O/action/replication_record"}},"created_at":"2026-07-02T01:18:31.961440+00:00","updated_at":"2026-07-02T01:18:31.961440+00:00"}