{
  "id": "1812.04925",
  "version": "v1",
  "published": "2018-12-12T12:56:23.000Z",
  "updated": "2018-12-12T12:56:23.000Z",
  "title": "On Bohr's theorem for general Dirichlet series",
  "authors": [
    "Ingo Schoolmann"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We present quantitative versions of Bohr's theorem on general Dirichlet series $D=\\sum a_{n} e^{-\\lambda_{n}s}$ assuming different assumptions on the frequency $\\lambda:=(\\lambda_{n})$, including the conditions introduced by Bohr and Landau. Therefore using the summation method by typical (first) means invented by M. Riesz, without any condition on $\\lambda$, we give upper bounds for the norm of the partial sum operator $S_{N}(D):=\\sum_{n=1}^{N} a_{n}(D)e^{-\\lambda_{n}s}$ of length $N$ on the space $\\mathcal{D}_{\\infty}^{ext}(\\lambda)$ of all somewhere convergent $\\lambda$-Dirichlet series allowing a holomorphic and bounded extension to the open right half plane $[Re>0]$. As a consequence for some classes of $\\lambda$'s we obtain a Montel theorem in $\\mathcal{D}_{\\infty}(\\lambda)$; the space of all $D \\in \\mathcal{D}_{\\infty}^{ext}(\\lambda)$ which converge on $[Re>0]$. Moreover following the ideas of Neder we give a construction of frequencies $\\lambda$ for which $\\mathcal{D}_{\\infty}(\\lambda)$ fails to be complete.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-12T12:56:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "general dirichlet series",
      "bohrs theorem",
      "open right half plane",
      "partial sum operator",
      "montel theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}