{"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"}