{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:MN35PRPFW27A5IZZURJQRRJQLO","short_pith_number":"pith:MN35PRPF","schema_version":"1.0","canonical_sha256":"6377d7c5e5b6be0ea339a45308c5305b8a015eb4274a2ae77be98cea1e26e748","source":{"kind":"arxiv","id":"1901.04837","version":6},"attestation_state":"computed","paper":{"title":"On some determinants involving the tangent function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2019-01-15T15:59:29Z","abstract_excerpt":"Let $p$ be an odd prime and let $a,b\\in\\mathbb Z$ with $p\\nmid ab$. In this paper we mainly evaluate $$T_p^{(\\delta)}(a,b,x):=\\det\\left[x+\\tan\\pi\\frac{aj^2+bk^2}p\\right]_{\\delta\\le j,k\\le (p-1)/2}\\ \\ (\\delta=0,1).$$ For example, in the case $p\\equiv3\\pmod4$ we show that $T_p^{(1)}(a,b,0)=0$ and $$T_p^{(0)}(a,b,x)=\\begin{cases} 2^{(p-1)/2}p^{(p+1)/4}&\\text{if}\\ (\\frac{ab}p)=1, \\\\p^{(p+1)/4}&\\text{if}\\ (\\frac{ab}p)=-1,\\end{cases}$$ where $(\\frac{\\cdot}p)$ is the Legendre symbol. When $(\\frac{-ab}p)=-1$, we also evaluate the determinant $\\det[x+\\cot\\pi\\frac{aj^2+bk^2}p]_{1\\le j,k\\le(p-1)/2}.$ In "},"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":"1901.04837","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-15T15:59:29Z","cross_cats_sorted":[],"title_canon_sha256":"fce341c4e00ac73e8b16e763d6392a2ae7bc7e8bcaaa4b53ade898c0fe944aba","abstract_canon_sha256":"e671c1a94d67063d27020fe300576c467cef6bb6fb1b463550b1069a745ac77a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:27:01.070308Z","signature_b64":"A7N9f8BLTP1yEtcAZdmWVuDnWPMlwI7HHoF5SpLxAAGAcTrX8yv+2BktvkhqGHkHrpGytCxO7h2QggtZthAXCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6377d7c5e5b6be0ea339a45308c5305b8a015eb4274a2ae77be98cea1e26e748","last_reissued_at":"2026-07-05T07:27:01.069827Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:27:01.069827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some determinants involving the tangent function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2019-01-15T15:59:29Z","abstract_excerpt":"Let $p$ be an odd prime and let $a,b\\in\\mathbb Z$ with $p\\nmid ab$. In this paper we mainly evaluate $$T_p^{(\\delta)}(a,b,x):=\\det\\left[x+\\tan\\pi\\frac{aj^2+bk^2}p\\right]_{\\delta\\le j,k\\le (p-1)/2}\\ \\ (\\delta=0,1).$$ For example, in the case $p\\equiv3\\pmod4$ we show that $T_p^{(1)}(a,b,0)=0$ and $$T_p^{(0)}(a,b,x)=\\begin{cases} 2^{(p-1)/2}p^{(p+1)/4}&\\text{if}\\ (\\frac{ab}p)=1, \\\\p^{(p+1)/4}&\\text{if}\\ (\\frac{ab}p)=-1,\\end{cases}$$ where $(\\frac{\\cdot}p)$ is the Legendre symbol. When $(\\frac{-ab}p)=-1$, we also evaluate the determinant $\\det[x+\\cot\\pi\\frac{aj^2+bk^2}p]_{1\\le j,k\\le(p-1)/2}.$ In "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.04837","kind":"arxiv","version":6},"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/1901.04837/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":"1901.04837","created_at":"2026-07-05T07:27:01.069881+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.04837v6","created_at":"2026-07-05T07:27:01.069881+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.04837","created_at":"2026-07-05T07:27:01.069881+00:00"},{"alias_kind":"pith_short_12","alias_value":"MN35PRPFW27A","created_at":"2026-07-05T07:27:01.069881+00:00"},{"alias_kind":"pith_short_16","alias_value":"MN35PRPFW27A5IZZ","created_at":"2026-07-05T07:27:01.069881+00:00"},{"alias_kind":"pith_short_8","alias_value":"MN35PRPF","created_at":"2026-07-05T07:27:01.069881+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/MN35PRPFW27A5IZZURJQRRJQLO","json":"https://pith.science/pith/MN35PRPFW27A5IZZURJQRRJQLO.json","graph_json":"https://pith.science/api/pith-number/MN35PRPFW27A5IZZURJQRRJQLO/graph.json","events_json":"https://pith.science/api/pith-number/MN35PRPFW27A5IZZURJQRRJQLO/events.json","paper":"https://pith.science/paper/MN35PRPF"},"agent_actions":{"view_html":"https://pith.science/pith/MN35PRPFW27A5IZZURJQRRJQLO","download_json":"https://pith.science/pith/MN35PRPFW27A5IZZURJQRRJQLO.json","view_paper":"https://pith.science/paper/MN35PRPF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.04837&json=true","fetch_graph":"https://pith.science/api/pith-number/MN35PRPFW27A5IZZURJQRRJQLO/graph.json","fetch_events":"https://pith.science/api/pith-number/MN35PRPFW27A5IZZURJQRRJQLO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MN35PRPFW27A5IZZURJQRRJQLO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MN35PRPFW27A5IZZURJQRRJQLO/action/storage_attestation","attest_author":"https://pith.science/pith/MN35PRPFW27A5IZZURJQRRJQLO/action/author_attestation","sign_citation":"https://pith.science/pith/MN35PRPFW27A5IZZURJQRRJQLO/action/citation_signature","submit_replication":"https://pith.science/pith/MN35PRPFW27A5IZZURJQRRJQLO/action/replication_record"}},"created_at":"2026-07-05T07:27:01.069881+00:00","updated_at":"2026-07-05T07:27:01.069881+00:00"}