{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:DCSR4QPFUGBAYQLTHSIHVQC3NK","short_pith_number":"pith:DCSR4QPF","schema_version":"1.0","canonical_sha256":"18a51e41e5a1820c41733c907ac05b6a9bc08e85f29ec219a87bb897567854d7","source":{"kind":"arxiv","id":"0807.0783","version":1},"attestation_state":"computed","paper":{"title":"Zeros of Dirichlet series with periodic coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas Weingartner, Eric Saias","submitted_at":"2008-07-04T15:48:39Z","abstract_excerpt":"Let $a=(a_n)_{n\\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of $\\sum_{n\\ge 1} a_n/n^s$, and $N_a(\\sigma_1, \\sigma_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $\\sigma_1 < \\Re s < \\sigma_2$, $|\\Im s | \\le T$. We extend previous results of Laurin\\v{c}ikas, Kaczorowski, Kulas, and Steuding, by showing that if $F_a(s)$ is not of the form $P(s) L_{\\chi} (s)$, where $P(s)$ is a Dirichlet polynomial and $L_{\\chi}(s)$ a Dirichlet L-function, then there exists an $\\eta=\\eta(a)>0$ such that for all $1/2 < \\sigma_1 < \\sigma_2 < 1+\\et"},"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":"0807.0783","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-07-04T15:48:39Z","cross_cats_sorted":[],"title_canon_sha256":"d459f48d0f3a407017aa90835befa9720c4103fdcca268b95804deaac3aec6e5","abstract_canon_sha256":"b2ff27e36c5dec4007c1c536cedb9578ab1875b088b359f482c351254bf8fe6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:15:45.604426Z","signature_b64":"NrCYl87Pl8+j0ZSAyLZH9IViQ4bClaCHbJl/UzrS10tMzq1CvMGtmlyTVodEr+IU26TyhjiTwuEGZ0yZz8joBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18a51e41e5a1820c41733c907ac05b6a9bc08e85f29ec219a87bb897567854d7","last_reissued_at":"2026-05-18T02:15:45.603725Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:15:45.603725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zeros of Dirichlet series with periodic coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas Weingartner, Eric Saias","submitted_at":"2008-07-04T15:48:39Z","abstract_excerpt":"Let $a=(a_n)_{n\\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of $\\sum_{n\\ge 1} a_n/n^s$, and $N_a(\\sigma_1, \\sigma_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $\\sigma_1 < \\Re s < \\sigma_2$, $|\\Im s | \\le T$. We extend previous results of Laurin\\v{c}ikas, Kaczorowski, Kulas, and Steuding, by showing that if $F_a(s)$ is not of the form $P(s) L_{\\chi} (s)$, where $P(s)$ is a Dirichlet polynomial and $L_{\\chi}(s)$ a Dirichlet L-function, then there exists an $\\eta=\\eta(a)>0$ such that for all $1/2 < \\sigma_1 < \\sigma_2 < 1+\\et"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.0783","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":"0807.0783","created_at":"2026-05-18T02:15:45.603820+00:00"},{"alias_kind":"arxiv_version","alias_value":"0807.0783v1","created_at":"2026-05-18T02:15:45.603820+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.0783","created_at":"2026-05-18T02:15:45.603820+00:00"},{"alias_kind":"pith_short_12","alias_value":"DCSR4QPFUGBA","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_16","alias_value":"DCSR4QPFUGBAYQLT","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_8","alias_value":"DCSR4QPF","created_at":"2026-05-18T12:25:57.157939+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/DCSR4QPFUGBAYQLTHSIHVQC3NK","json":"https://pith.science/pith/DCSR4QPFUGBAYQLTHSIHVQC3NK.json","graph_json":"https://pith.science/api/pith-number/DCSR4QPFUGBAYQLTHSIHVQC3NK/graph.json","events_json":"https://pith.science/api/pith-number/DCSR4QPFUGBAYQLTHSIHVQC3NK/events.json","paper":"https://pith.science/paper/DCSR4QPF"},"agent_actions":{"view_html":"https://pith.science/pith/DCSR4QPFUGBAYQLTHSIHVQC3NK","download_json":"https://pith.science/pith/DCSR4QPFUGBAYQLTHSIHVQC3NK.json","view_paper":"https://pith.science/paper/DCSR4QPF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0807.0783&json=true","fetch_graph":"https://pith.science/api/pith-number/DCSR4QPFUGBAYQLTHSIHVQC3NK/graph.json","fetch_events":"https://pith.science/api/pith-number/DCSR4QPFUGBAYQLTHSIHVQC3NK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DCSR4QPFUGBAYQLTHSIHVQC3NK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DCSR4QPFUGBAYQLTHSIHVQC3NK/action/storage_attestation","attest_author":"https://pith.science/pith/DCSR4QPFUGBAYQLTHSIHVQC3NK/action/author_attestation","sign_citation":"https://pith.science/pith/DCSR4QPFUGBAYQLTHSIHVQC3NK/action/citation_signature","submit_replication":"https://pith.science/pith/DCSR4QPFUGBAYQLTHSIHVQC3NK/action/replication_record"}},"created_at":"2026-05-18T02:15:45.603820+00:00","updated_at":"2026-05-18T02:15:45.603820+00:00"}