{
  "id": "1901.04837",
  "version": "v1",
  "published": "2019-01-15T15:59:29.000Z",
  "updated": "2019-01-15T15:59:29.000Z",
  "title": "On some determinants involving the tangent function",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be an odd prime and let $a,b\\in\\mathbb Z$ with $p\\nmid ab$. In this paper we mainly evaluate $$T_p^{(\\delta)}(a,b):=\\det\\left[\\tan\\pi\\frac{aj^2+bk^2}p\\right]_{\\delta\\le j,k\\le (p-1)/2}\\ \\ (\\delta=0,1).$$ For example, in the case $p\\equiv3\\pmod4$ we show that $T_p^{(1)}(a,b)=0$ and $$T_p^{(0)}(a,b)=\\begin{cases} 2^{(p-1)/2}p^{(p+1)/4}&\\text{if}\\ (\\frac{ab}p)=1, \\\\p^{(p+1)/4}&\\text{if}\\ (\\frac{ab}p)=-1.\\end{cases}$$ When $(\\frac{-ab}p)=-1$, we also evaluate the determinant $\\det[\\cot\\pi\\frac{aj^2+bk^2}p]_{1\\le j,k\\le(p-1)/2}.$ We also pose several conjectures one of which states that the class number of the quadratic field $\\mathbb Q(\\sqrt{p^*})$ with $p^*=(-1)^{(p-1)/2}p$ is equal to $$\\left(\\frac{-2}p\\right)2^{-(p-3)/2}p^{-(p-5)/4}\\det\\left[\\cot\\pi\\frac{jk}p\\right]_{1\\ls j,k\\ls (p-1)/2}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-15T15:59:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11C20",
      "33B10",
      "11A15",
      "15A99"
    ],
    "keywords": [
      "tangent function",
      "determinant",
      "odd prime",
      "class number",
      "quadratic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}