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We develop a new approach to zero-free regions for $L$-functions via lower bounds for power sums, proving for all $\\varepsilon>0$ the existence of ineffective constants $c=c_{n,F,\\varepsilon}>0$ and $c'=c'_{n,F,\\pi',\\varepsilon}>0$ such that the standard $L$-function $L(s,\\pi)$ satisfies \\[ |L(\\sigma+it,\\pi)|\\geq c(\\mathfrak{C}_{\\pi}(|t|+3))^{-\\varepsilon},\\qquad \\sigma\\geq 1-c(\\mathfrak{C}_{\\pi}(|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":"2608.12257","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-08-12T17:00:49Z","cross_cats_sorted":[],"title_canon_sha256":"805d836b6eef54e8bcf0021ec0142a665f4641e2fd360c5e9098085e51704f9b","abstract_canon_sha256":"441ac0378d20d8e5b16b26e5d1ffa71df9c6e84b7654e32ff3c40f8df30e431c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-13T01:30:38.284281Z","signature_b64":"KsGgXNwdgijK7XahaxADPBa/84EfM1qnEmEeOnQzzMBngwslpGf9iNN2QG6N3J6y4YPpm5xY8GasYAHxXcFHAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"875536b97854a7e2fbd5a31893c712c75e5e8a074daed52407006096fc9d76b8","last_reissued_at":"2026-08-13T01:30:38.281941Z","signature_status":"signed_v1","first_computed_at":"2026-08-13T01:30:38.281941Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Power sums and Siegel-type zero-free regions for L-functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jesse Thorner","submitted_at":"2026-08-12T17:00:49Z","abstract_excerpt":"Let $\\pi$ and $\\pi'$ be unitary cuspidal automorphic representations of $\\mathrm{GL}(n)$ and $\\mathrm{GL}(n')$ over a number field $F$. Let $\\mathfrak{C}_{\\pi}$ be the analytic conductor of $\\pi$. We develop a new approach to zero-free regions for $L$-functions via lower bounds for power sums, proving for all $\\varepsilon>0$ the existence of ineffective constants $c=c_{n,F,\\varepsilon}>0$ and $c'=c'_{n,F,\\pi',\\varepsilon}>0$ such that the standard $L$-function $L(s,\\pi)$ satisfies \\[ |L(\\sigma+it,\\pi)|\\geq c(\\mathfrak{C}_{\\pi}(|t|+3))^{-\\varepsilon},\\qquad \\sigma\\geq 1-c(\\mathfrak{C}_{\\pi}(|t|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.12257","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/2608.12257/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":"2608.12257","created_at":"2026-08-13T01:30:38.283063+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.12257v1","created_at":"2026-08-13T01:30:38.283063+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.12257","created_at":"2026-08-13T01:30:38.283063+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q5KTNOLYKST6","created_at":"2026-08-13T01:30:38.283063+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q5KTNOLYKST6F66V","created_at":"2026-08-13T01:30:38.283063+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q5KTNOLY","created_at":"2026-08-13T01:30:38.283063+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/Q5KTNOLYKST6F66VUMMJHRYSY5","json":"https://pith.science/pith/Q5KTNOLYKST6F66VUMMJHRYSY5.json","graph_json":"https://pith.science/api/pith-number/Q5KTNOLYKST6F66VUMMJHRYSY5/graph.json","events_json":"https://pith.science/api/pith-number/Q5KTNOLYKST6F66VUMMJHRYSY5/events.json","paper":"https://pith.science/paper/Q5KTNOLY"},"agent_actions":{"view_html":"https://pith.science/pith/Q5KTNOLYKST6F66VUMMJHRYSY5","download_json":"https://pith.science/pith/Q5KTNOLYKST6F66VUMMJHRYSY5.json","view_paper":"https://pith.science/paper/Q5KTNOLY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.12257&json=true","fetch_graph":"https://pith.science/api/pith-number/Q5KTNOLYKST6F66VUMMJHRYSY5/graph.json","fetch_events":"https://pith.science/api/pith-number/Q5KTNOLYKST6F66VUMMJHRYSY5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q5KTNOLYKST6F66VUMMJHRYSY5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q5KTNOLYKST6F66VUMMJHRYSY5/action/storage_attestation","attest_author":"https://pith.science/pith/Q5KTNOLYKST6F66VUMMJHRYSY5/action/author_attestation","sign_citation":"https://pith.science/pith/Q5KTNOLYKST6F66VUMMJHRYSY5/action/citation_signature","submit_replication":"https://pith.science/pith/Q5KTNOLYKST6F66VUMMJHRYSY5/action/replication_record"}},"created_at":"2026-08-13T01:30:38.283063+00:00","updated_at":"2026-08-13T01:30:38.283063+00:00"}