{
  "id": "2505.10341",
  "version": "v1",
  "published": "2025-05-15T14:29:14.000Z",
  "updated": "2025-05-15T14:29:14.000Z",
  "title": "A new result on the divisor problem in arithmetic progressions modulo a prime power",
  "authors": [
    "Mingxuan Zhong",
    "Tianping Zhang"
  ],
  "comment": "19 pages, accepted by SCIENTIA SINICA Mathematica (in Chinese)",
  "doi": "10.1360/SCM-2024-0717",
  "categories": [
    "math.NT"
  ],
  "abstract": "We derive an asymptotic formula for the divisor function $\\tau(k)$ in an arithmetic progression $k\\equiv a(\\bmod \\ q)$, uniformly for $q\\leq X^{\\Delta_{n,l}}$ with $(q,a)=1$. The parameter $\\Delta_{n,l}$ is defined as $$ \\Delta_{n,l}=\\frac{1-\\frac{3}{2^{2^l+2l-3}}}{1-\\frac{1}{n2^{l-1}}}. $$ Specifically, by setting $l=2$, we achieve $\\Delta_{n,l}>3/4+5/32$, which surpasses the result obtained by Liu, Shparlinski, and Zhang (2018). Meanwhile, this has also improved upon the result of Wu and Xi (2021). Notably, Hooley, Linnik, and Selberg independently established that the asymptotic formula holds for $q\\leq X^{2/3-\\varepsilon}$. Irving was the first to surpass the $2/3-$barrier for certain special moduli. We break the classical $3/4-$barrier in the case of prime power moduli and extend the range of $q$. Our main ingredients borrow from Mangerel's adaptation of Mili\\'{c}evi\\'{c} and Zhang's methodology in dealing with a specific class of weighted Kloosterman sums, rather than adopting Korobov's technique employed by Liu, Shparlinski, and Zhang.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-15T14:29:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic progressions modulo",
      "divisor problem",
      "asymptotic formula holds",
      "main ingredients borrow",
      "prime power moduli"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "zh",
      "license": "arXiv",
      "status": "editable"
    }
  }
}