{
  "id": "2410.12305",
  "version": "v1",
  "published": "2024-10-16T07:14:49.000Z",
  "updated": "2024-10-16T07:14:49.000Z",
  "title": "Sums of Fourier coefficients involving theta series and Dirichlet characters",
  "authors": [
    "Yanxue Yu"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a holomorphic or Maass cusp forms for $ \\rm SL_2(\\mathbb{Z})$ with normalized Fourier coefficients $\\lambda_f(n)$ and \\bna r_{\\ell}(n)=\\#\\left\\{(n_1,\\cdots,n_{\\ell})\\in \\mathbb{Z}^2:n_1^2+\\cdots+n_{\\ell}^2=n\\right\\}. \\ena Let $\\chi$ be a primitive Dirichlet character of modulus $p$, a prime. In this paper, we are concerned with obtaining nontrivial estimates for the sum \\bna \\sum_{n\\geq1}\\lambda_f(n)r_{\\ell}(n)\\chi(n)w\\left(\\frac{n}{X}\\right) \\ena for any $\\ell \\geq 3$, where $w(x)$ be a smooth function compactly supported in $[1/2,1]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-16T07:14:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "theta series",
      "maass cusp forms",
      "primitive dirichlet character",
      "obtaining nontrivial estimates",
      "smooth function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}