{
  "abstract": "Denote by $\\mathrm{Kl}(a,n)$ the normalized Kloosterman sum modulo $n$. In this paper, we study the moments of $|\\mathrm{Kl}(1,n)|$ on average over $n\\leqslant X$, and show that \\begin{align*} \\sum_{n\\leqslant X}|\\mathrm{Kl}(1,n)|\\asymp X(\\log X)^{\\frac{8}{3π}-1}. \\end{align*} The lower bound is proven by considering a subfamily of moduli which factorize in a nice way such that vertical Sato--Tate distributions of Kloosterman sums (as variants of Katz) apply. The upper bound is based on a new density version of upper bound sieves, combining with averages of divisor functions in arithmetic progressions and a similar vertical Sato--Tate distribution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11083",
  "author_total": 1,
  "authors": [
    {
      "name": "Ping Xi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ba18qy57h4x5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11083",
  "fulltext_url": null,
  "id": "9ba18qy57h4x5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11083",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221145035",
  "pdf_url": "https://arxiv.org/pdf/2610.11083",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ba18qy57h4x5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11083",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11083"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Horizontal moments of Kloosterman sums and a density version of sieves",
  "title_text": "Horizontal moments of Kloosterman sums and a density version of sieves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/9ba18qy57h4x5",
  "venue": "arXiv",
  "versions": []
}