{
  "id": "2104.02711",
  "version": "v1",
  "published": "2021-04-06T17:59:58.000Z",
  "updated": "2021-04-06T17:59:58.000Z",
  "title": "A Bombieri-Vinogradov theorem for higher rank groups",
  "authors": [
    "Yujiao Jiang",
    "Guangshi Lü",
    "Jesse Thorner",
    "Zihao Wang"
  ],
  "comment": "38 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We establish a result of Bombieri-Vinogradov type for the Dirichlet coefficients at prime ideals of the standard $L$-function associated to a self-dual cuspidal automorphic representation $\\pi$ of $\\mathrm{GL}_n$ over a number field $F$ which is not a quadratic twist of itself. Our result does not rely on any unproven progress towards the generalized Ramanujan conjecture or the nonexistence of Landau-Siegel zeros. In particular, when $\\pi$ is fixed and not equal to a quadratic twist of itself, we prove the first unconditional Siegel-type lower bound for the twisted $L$-values $|L(1,\\pi\\otimes\\chi)|$ in the $\\chi$-aspect, where $\\chi$ is a primitive quadratic Hecke character over $F$. Our result improves the levels of distribution in other works that relied on these unproven hypotheses. As applications, when $n=2,3,4$, we prove a $\\mathrm{GL}_n$ analogue of the Titchmarsh divisor problem and a nontrivial bound for a certain $\\mathrm{GL}_n\\times\\mathrm{GL}_2$ shifted convolution sum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-06T17:59:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher rank groups",
      "bombieri-vinogradov theorem",
      "first unconditional siegel-type lower bound",
      "self-dual cuspidal automorphic representation",
      "quadratic twist"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}