{
  "abstract": "Let x be the smallest x > 1 with |M (x)| ≥ √\n\nx, where M (x) = ∑︁ n≤x µ(n). We prove that\n\nx < exp(3 484 971 332 733 178 124) < exp(3.485 × 10 18 ), improving the bound x < exp(1.96 × 10 19 ) of Kim and Nguyen by a factor of about 5.6 in the exponent. The proof follows the effective method of Pintz, as refined by Saouter and te Riele. The criterion gives a counterexample near e u , or earlier, when the absolute value of a Gaussian-weighted finite zero sum exceeds 1 by more than an explicit error. In the local case, the sum is compared with a positive Gaussian average of M (x)/ √ x, with both contour and smoothing errors controlled. The smoothing parameter k in the weight e -kγ 2 controls a trade-off: smaller k gives more weight to higher zeros but requires a higher cutoff to keep the error small. All previous records used k ≥ 3 • 10 -9 . We re-derive the error terms for general parameters and work at k = 1.5 • 10 -10 , at a point found by lattice-based search. This requires the sum over all 2 325 572 zeros of height below 1.3 • 10 6 , which we evaluate in rigorous ball arithmetic. We also recompute, from certified zeros, the bound on 1/ζ required on the relevant height interval.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Zhenyu Bi"
    },
    {
      "name": "Enze Hou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7yhmvfqd3ztq8.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "7yhmvfqd3ztq8",
  "issns": [],
  "landing_url": "https://hal.science/hal-05838582v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-nt"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05838582/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7yhmvfqd3ztq8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05838582",
      "source": "hal",
      "url": "https://hal.science/hal-05838582v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-nt",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "AN IMPROVED UPPER BOUND FOR THE SMALLEST COUNTEREXAMPLE TO THE MERTENS CONJECTURE",
  "title_text": "AN IMPROVED UPPER BOUND FOR THE SMALLEST COUNTEREXAMPLE TO THE MERTENS CONJECTURE",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:01Z",
  "url": "https://research.draconitic.com/paper/7yhmvfqd3ztq8",
  "venue": "HAL",
  "versions": []
}