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Let $d$ be the degree of $F(s)$. We show that $\\sum_{n<X}|a_n|=\\Omega(X^{\\frac{1}{2}+\\frac{1}{2d}})$, and hence, that the abscissa of absolute convergence of $\\sigma_a$ of $F(s)$ must satisfy $\\sigma_a\\ge 1/2+1/2d$."},"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":"2105.06957","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-05-14T16:57:48Z","cross_cats_sorted":[],"title_canon_sha256":"df6d3b80a061582d9710758519d87cf809ddfd965aedb5e9c252e6d960c8a460","abstract_canon_sha256":"bc66161ed7638a4977f0cd47404bc862f12228b6980fa46491a0b6d6fee46374"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:40:23.292464Z","signature_b64":"8Db497cDjWOANfE8hyQsznvYZGtH4Po4QfvvVibq16aOs3lUDAI/zcS7NwLz4X4tmZFP7nmqw6F/4GxL7U0iCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16c9c7c3e307c12254c21e5ecef4a663721200e27cd2493c6ec879218b101544","last_reissued_at":"2026-07-05T02:40:23.292027Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:40:23.292027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the absolute convergence of automorphic Dirichlet series","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ravi Raghunathan","submitted_at":"2021-05-14T16:57:48Z","abstract_excerpt":"Let $F(s)=\\sum_{n=1}^{\\infty}\\frac{a_n}{n^s}$ be a Dirichlet series in the axiomatically defined class ${\\mathfrak A}^{\\#}$ . The class ${\\mathfrak A}^{\\#}$ is known to contain the extended Selberg class ${\\mathcal S}^{\\#}$, as well as all the $L$-functions of automorphic forms on $GL_n/K$, where $K$ is a number field. Let $d$ be the degree of $F(s)$. We show that $\\sum_{n<X}|a_n|=\\Omega(X^{\\frac{1}{2}+\\frac{1}{2d}})$, and hence, that the abscissa of absolute convergence of $\\sigma_a$ of $F(s)$ must satisfy $\\sigma_a\\ge 1/2+1/2d$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.06957","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/2105.06957/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":"2105.06957","created_at":"2026-07-05T02:40:23.292093+00:00"},{"alias_kind":"arxiv_version","alias_value":"2105.06957v1","created_at":"2026-07-05T02:40:23.292093+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.06957","created_at":"2026-07-05T02:40:23.292093+00:00"},{"alias_kind":"pith_short_12","alias_value":"C3E4PQ7DA7AS","created_at":"2026-07-05T02:40:23.292093+00:00"},{"alias_kind":"pith_short_16","alias_value":"C3E4PQ7DA7ASEVGC","created_at":"2026-07-05T02:40:23.292093+00:00"},{"alias_kind":"pith_short_8","alias_value":"C3E4PQ7D","created_at":"2026-07-05T02:40:23.292093+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/C3E4PQ7DA7ASEVGCDZPM55FGMN","json":"https://pith.science/pith/C3E4PQ7DA7ASEVGCDZPM55FGMN.json","graph_json":"https://pith.science/api/pith-number/C3E4PQ7DA7ASEVGCDZPM55FGMN/graph.json","events_json":"https://pith.science/api/pith-number/C3E4PQ7DA7ASEVGCDZPM55FGMN/events.json","paper":"https://pith.science/paper/C3E4PQ7D"},"agent_actions":{"view_html":"https://pith.science/pith/C3E4PQ7DA7ASEVGCDZPM55FGMN","download_json":"https://pith.science/pith/C3E4PQ7DA7ASEVGCDZPM55FGMN.json","view_paper":"https://pith.science/paper/C3E4PQ7D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2105.06957&json=true","fetch_graph":"https://pith.science/api/pith-number/C3E4PQ7DA7ASEVGCDZPM55FGMN/graph.json","fetch_events":"https://pith.science/api/pith-number/C3E4PQ7DA7ASEVGCDZPM55FGMN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C3E4PQ7DA7ASEVGCDZPM55FGMN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C3E4PQ7DA7ASEVGCDZPM55FGMN/action/storage_attestation","attest_author":"https://pith.science/pith/C3E4PQ7DA7ASEVGCDZPM55FGMN/action/author_attestation","sign_citation":"https://pith.science/pith/C3E4PQ7DA7ASEVGCDZPM55FGMN/action/citation_signature","submit_replication":"https://pith.science/pith/C3E4PQ7DA7ASEVGCDZPM55FGMN/action/replication_record"}},"created_at":"2026-07-05T02:40:23.292093+00:00","updated_at":"2026-07-05T02:40:23.292093+00:00"}