{
  "id": "2409.18537",
  "version": "v1",
  "published": "2024-09-27T08:15:25.000Z",
  "updated": "2024-09-27T08:15:25.000Z",
  "title": "Transcendence of values of logarithms of $E$-functions",
  "authors": [
    "Stéphane Fischler",
    "Tanguy Rivoal"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be an $E$-function (in Siegel's sense) not of the form $e^{\\beta z}$, $\\beta \\in \\overline{\\mathbb{Q}}$, and let $\\log$ denote any fixed determination of the complex logarithm. We first prove that there exists a finite set $S(f)$ such that for all $\\xi\\in \\overline{\\mathbb{Q}}\\setminus S(f)$, $\\log(f(\\xi))$ is a transcendental number. We then quantify this result when $f$ is an $E$-function in the strict sense with rational coefficients, by proving an irrationality measure of $\\ln(f(\\xi))$ when $\\xi\\in \\mathbb{Q}\\setminus S(f)$ and $f(\\xi)\\gt0$. This measure implies that $\\ln(f(\\xi))$ is not an ultra-Liouville number, as defined by Marques and Moreira. The proof of our first result, which is in fact more general, uses in particular a recent theorem of Delaygue. The proof of the second result, which is independent of the first one, is a consequence of a new linear independence measure for values of linearly independent $E$-functions in the strict sense with rational coefficients, where emphasis is put on other parameters than on the height, contrary to the case in Shidlovskii's classical measure for instance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-27T08:15:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational coefficients",
      "transcendence",
      "strict sense",
      "linear independence measure",
      "measure implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}