{
  "abstract": "For real $t$ and $x\\ge 1$, set \\[ S(x,t)=\\sum_{n\\le x} n^{\\ii t}. \\] We prove an unconditional inverse theorem relating large values of $S(x,t)$, with $|t|$ large, to zeros of the Riemann zeta function near height $t$. More precisely, if $T\\le |t|\\le 2T$, $\\exp(\\sqrt{\\log T})\\le x\\le \\sqrt T$, and $|S(x,t)|=x/N$ with $N\\le (\\log x)^{1/100}$, then for every $cN^6\\le L\\le (\\log x)/2$ a disk centered at $1+\\iiϕ$, where $|ϕ-t|\\ll N$, contains at least $L/360$ zeros of $ζ(s)$. As a consequence, a quantitative restriction on zeros in a short family of arbitrarily thin fixed windows to the left of the line $\\Ree s=1$ yields $S(x,t)\\ll x/(\\log x)^{1/100}$ in polynomial ranges of $x$. The proof adapts the zero-forcing mechanism of Granville and Soundararajan for large character sums. In the zeta setting the spectral height is shifted by $t$, and an additional residue from the pole of $ζ(s)$ appears in the Gaussian transform; in the range considered here that residue is exponentially small.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.31060",
  "author_total": 4,
  "authors": [
    {
      "name": "Zikang Dong"
    },
    {
      "name": "Ruihua Wang"
    },
    {
      "name": "Weijia Wang"
    },
    {
      "name": "Hao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/czznnvb0mgpnk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.31060",
  "fulltext_url": null,
  "id": "czznnvb0mgpnk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.31060",
  "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": "W7204899929",
  "pdf_url": "https://arxiv.org/pdf/2608.31060",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/czznnvb0mgpnk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.31060",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.31060"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Large zeta sums and zeros of the Riemann zeta function",
  "title_text": "Large zeta sums and zeros of the Riemann zeta function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/czznnvb0mgpnk",
  "venue": "arXiv",
  "versions": []
}