{
  "abstract": "This research report gives rigorous, computer-assisted proofs of several explicit results on oscillations of arithmetic\n\nsums, each with a checkable certificate and a verification path that does not depend on the author's software.\n\nMain results: (i) an explicit region where the error term of Mertens' first theorem is negative (problem 14 of the list of\n\nHamieh, Kadiri, Martin and Ng); (ii) a sign change in the prime race 24k+1 versus 24k+11 below 2.79·10^48, improving the\n\nbound 10^353 of Ford and Hudson; (iii) an explicit negative value of Shanks' sum Σ λ(n)χ_{−4}(n) and an unconditional\n\nproof, by the classical method of Landau and Ingham, that it changes sign infinitely often; (iv) explicit counterexamples\n\nto five strict inequalities of Conjecture 1.3 of Z.-W. Sun, explicit failures of his Hypothesis 1.2 for Q(i) and Q(√−2),\n\nan explicit height for violations of the two upper bounds of his Hypothesis 1.1 (already known to fail infinitely often,\n\nby Mossinghoff and Trudgian), and an unconditional proof, by Selberg–Delange asymptotics, that\n\neach of the seventeen eventual inequalities of his Conjecture 1.2 fails infinitely often; (v) three explicit regions with\n\nπ(x) > li(x), near 10^3127.6, 10^6217.1 and 10^8045.6; (vi) under GRH, a complete classification of the pairs (q,a) for\n\nwhich the integrated prime race ∫(φ(q)π(t;q,a) − li t)dt is eventually negative (PIMS problem 23; LI needed only for the\n\n\"only if\" direction), with a certified correction of one omission in Table 1 of Zhao (2025).\n\nAll numerical claims used in proofs are backed by ball (interval) arithmetic with every error term bounded. The\n\naccompanying package contains the certificates, short checkers (Python + python-flint/Arb), logs and SHA-256 sums; the\n\nreport lists, for every result, what can be checked by hand and what requires a computer. The computations and the\n\ndrafting involved computer-assisted tools, including AI-based tools; the author takes full responsibility for all\n\nmathematical statements. The author is an independent researcher; novelty statements mean \"not found in the literature\n\nconsulted\", and corrections are welcome.\n\nVersion 1.2 (October 2026). (i) A statement about another paper is withdrawn: versions 1.0 and 1.1 described a local\n\nfactor in (4.1)/(4.13) of Humphries, Shekatkar and Wong (2019) as having the wrong sign; that description was incorrect.\n\nTheir formulas are correct for the function of their Definition 1.2 (for q = 4 this is λ(n;4,1,2), for which their constant\n\nc of (4.13) equals 0.4387); the constant 2.5567 belongs to the related function λ(n;4,1) of their Figure 1 and computations,\n\nand the paper uses the word \"complement\" in these two senses. Appendix K now describes both functions. (ii) The DOI of one\n\nreference is corrected, and a remark about a 2025 preprint is removed. (iii) Earlier work is acknowledged more precisely:\n\nTable 1 of Büthe (2015) already lists the height of Theorem 1 as a numerical candidate for Mertens' second theorem\n\n(Theorem 1 certifies a sign change for the first theorem there), and Mossinghoff and Trudgian (2021, §6) suggested studying\n\nSun's conjectures for S_{−4} through weak independence of zeros; the attribution of the method of Theorems 8 and J.4 is\n\nmade precise. (iv) A claim in Appendix L (that some cases of Theorem 7 can be checked by hand) is corrected. (v) Several\n\ndisplayed values are corrected: the second interval of Theorem 6 is now printed rounded outwards; supporting bounds and\n\nratios are rounded outwards; several ledgers now give the actual centre and radius of their enclosures (Theorems 1, 2, 8\n\nand J.4); exponents and subscripts of log that were printed on the line in some formulas are now typeset correctly, and the\n\nquotation of Lehman's theorem in Appendix E, whose last term ran past the margin, is now printed in full.\n\n(vi) The certificate package adds an exact sieve, an independent check of the values up to 10^11 for A(x) and S_{−3}, and\n\na guide to verifying every result with public software only. Apart from the withdrawn statement about Humphries–Shekatkar–\n\nWong (Appendix K), no theorem, set, sign or region of a result of the report changes (one printed interval of Theorem 6 is\n\nnow rounded outwards), and no certificate data file changes. The complete list is in the section \"Changes in version 1.2\"\n\nof the report.",
  "abstract_shown": true,
  "affiliations": [
    "Independent researcher"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Leandro Hector Gallego",
      "orcid": "0009-0006-8285-3657"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3v8b03m80efcj.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23597138",
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  "id": "3v8b03m80efcj",
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  "landing_url": "https://zenodo.org/records/23597138",
  "language": "eng",
  "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": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23597138/files/Gallego_2026_explicit_sign_changes_EN_v1.2.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3v8b03m80efcj.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T12:21:09Z",
      "native_id": "23597138",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23597138"
    }
  ],
  "subjects": [],
  "title": "Explicit sign changes in comparative prime number theory: computer-assisted proofs with checkable certificates",
  "title_text": "Explicit sign changes in comparative prime number theory: computer-assisted proofs with checkable certificates",
  "type": "preprint",
  "updated_at": "2026-11-06T12:21:09Z",
  "url": "https://research.draconitic.com/paper/3v8b03m80efcj",
  "venue": "Zenodo",
  "versions": []
}