Explicit sign changes in comparative prime number theory: computer-assisted proofs with checkable certificates
- 1Independent researcher
Abstract
This research report gives rigorous, computer-assisted proofs of several explicit results on oscillations of arithmetic
sums, each with a checkable certificate and a verification path that does not depend on the author's software.
Main results: (i) an explicit region where the error term of Mertens' first theorem is negative (problem 14 of the list of
Hamieh, Kadiri, Martin and Ng); (ii) a sign change in the prime race 24k+1 versus 24k+11 below 2.79·10^48, improving the
bound 10^353 of Ford and Hudson; (iii) an explicit negative value of Shanks' sum Σ λ(n)χ_{−4}(n) and an unconditional
proof, by the classical method of Landau and Ingham, that it changes sign infinitely often; (iv) explicit counterexamples
to five strict inequalities of Conjecture 1.3 of Z.-W. Sun, explicit failures of his Hypothesis 1.2 for Q(i) and Q(√−2),
an explicit height for violations of the two upper bounds of his Hypothesis 1.1 (already known to fail infinitely often,
by Mossinghoff and Trudgian), and an unconditional proof, by Selberg–Delange asymptotics, that
each of the seventeen eventual inequalities of his Conjecture 1.2 fails infinitely often; (v) three explicit regions with
π(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
which the integrated prime race ∫(φ(q)π(t;q,a) − li t)dt is eventually negative (PIMS problem 23; LI needed only for the
"only if" direction), with a certified correction of one omission in Table 1 of Zhao (2025).
All numerical claims used in proofs are backed by ball (interval) arithmetic with every error term bounded. The
accompanying package contains the certificates, short checkers (Python + python-flint/Arb), logs and SHA-256 sums; the
report lists, for every result, what can be checked by hand and what requires a computer. The computations and the
drafting involved computer-assisted tools, including AI-based tools; the author takes full responsibility for all
mathematical statements. The author is an independent researcher; novelty statements mean "not found in the literature
consulted", and corrections are welcome.
Version 1.2 (October 2026). (i) A statement about another paper is withdrawn: versions 1.0 and 1.1 described a local
factor in (4.1)/(4.13) of Humphries, Shekatkar and Wong (2019) as having the wrong sign; that description was incorrect.
Their 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
c 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,
and the paper uses the word "complement" in these two senses. Appendix K now describes both functions. (ii) The DOI of one
reference is corrected, and a remark about a 2025 preprint is removed. (iii) Earlier work is acknowledged more precisely:
Table 1 of Büthe (2015) already lists the height of Theorem 1 as a numerical candidate for Mertens' second theorem
(Theorem 1 certifies a sign change for the first theorem there), and Mossinghoff and Trudgian (2021, §6) suggested studying
Sun's conjectures for S_{−4} through weak independence of zeros; the attribution of the method of Theorems 8 and J.4 is
made precise. (iv) A claim in Appendix L (that some cases of Theorem 7 can be checked by hand) is corrected. (v) Several
displayed values are corrected: the second interval of Theorem 6 is now printed rounded outwards; supporting bounds and
ratios are rounded outwards; several ledgers now give the actual centre and radius of their enclosures (Theorems 1, 2, 8
and J.4); exponents and subscripts of log that were printed on the line in some formulas are now typeset correctly, and the
quotation of Lehman's theorem in Appendix E, whose last term ran past the margin, is now printed in full.
(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
a guide to verifying every result with public software only. Apart from the withdrawn statement about Humphries–Shekatkar–
Wong (Appendix K), no theorem, set, sign or region of a result of the report changes (one printed interval of Theorem 6 is
now rounded outwards), and no certificate data file changes. The complete list is in the section "Changes in version 1.2"
of the report.
Related
- Ordered-weight certificates for odd cycles in dense coprime graphs: A computer-assisted argument for Erdős Problem 883(i)
- Explicit near-D5 local results for the five-dimensional kissing number (computer-assisted, unreviewed)
- Algebraic extinction in acyclic spreading models: explicit counterexamples, genericity and support-checkable repairs
- Optimizing Explicit Unit-Distance Lower-Bound Certificates
- Independently Checkable Certificates for Connected DC Optimal Transmission Switching
BibTeX
@misc{gallego2026explicit,
title = {{Explicit sign changes in comparative prime number theory: computer-assisted proofs with checkable certificates}},
author = {Leandro Hector Gallego},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23597138},
url = {https://zenodo.org/records/23597138},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Explicit sign changes in comparative prime number theory: computer-assisted proofs with checkable certificates
AU - Leandro Hector Gallego
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23597138
UR - https://zenodo.org/records/23597138
AB - This research report gives rigorous, computer-assisted proofs of several explicit results on oscillations of arithmetic sums, each with a checkable certificate and a verification path that does not depend on the author's software. Main results: (i) an explicit region where the error term of Mertens' first theorem is negative (problem 14 of the list of Hamieh, Kadiri, Martin and Ng); (ii) a sign change in the prime race 24k+1 versus 24k+11 below 2.79·10^48, improving the bound 10^353 of Ford and Hudson; (iii) an explicit negative value of Shanks' sum Σ λ(n)χ_{−4}(n) and an unconditional proof, by the classical method of Landau and Ingham, that it changes sign infinitely often; (iv) explicit counterexamples to five strict inequalities of Conjecture 1.3 of Z.-W. Sun, explicit failures of his Hypothesis 1.2 for Q(i) and Q(√−2), an explicit height for violations of the two upper bounds of his Hypothesis 1.1 (already known to fail infinitely often, by Mossinghoff and Trudgian), and an unconditional proof, by Selberg–Delange asymptotics, that each of the seventeen eventual inequalities of his Conjecture 1.2 fails infinitely often; (v) three explicit regions with π(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 which the integrated prime race ∫(φ(q)π(t;q,a) − li t)dt is eventually negative (PIMS problem 23; LI needed only for the "only if" direction), with a certified correction of one omission in Table 1 of Zhao (2025). All numerical claims used in proofs are backed by ball (interval) arithmetic with every error term bounded. The accompanying package contains the certificates, short checkers (Python + python-flint/Arb), logs and SHA-256 sums; the report lists, for every result, what can be checked by hand and what requires a computer. The computations and the drafting involved computer-assisted tools, including AI-based tools; the author takes full responsibility for all mathematical statements. The author is an independent researcher; novelty statements mean "not found in the literature consulted", and corrections are welcome. Version 1.2 (October 2026). (i) A statement about another paper is withdrawn: versions 1.0 and 1.1 described a local factor in (4.1)/(4.13) of Humphries, Shekatkar and Wong (2019) as having the wrong sign; that description was incorrect. Their 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 c 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, and the paper uses the word "complement" in these two senses. Appendix K now describes both functions. (ii) The DOI of one reference is corrected, and a remark about a 2025 preprint is removed. (iii) Earlier work is acknowledged more precisely: Table 1 of Büthe (2015) already lists the height of Theorem 1 as a numerical candidate for Mertens' second theorem (Theorem 1 certifies a sign change for the first theorem there), and Mossinghoff and Trudgian (2021, §6) suggested studying Sun's conjectures for S_{−4} through weak independence of zeros; the attribution of the method of Theorems 8 and J.4 is made precise. (iv) A claim in Appendix L (that some cases of Theorem 7 can be checked by hand) is corrected. (v) Several displayed values are corrected: the second interval of Theorem 6 is now printed rounded outwards; supporting bounds and ratios are rounded outwards; several ledgers now give the actual centre and radius of their enclosures (Theorems 1, 2, 8 and J.4); exponents and subscripts of log that were printed on the line in some formulas are now typeset correctly, and the quotation of Lehman's theorem in Appendix E, whose last term ran past the margin, is now printed in full. (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 a guide to verifying every result with public software only. Apart from the withdrawn statement about Humphries–Shekatkar– Wong (Appendix K), no theorem, set, sign or region of a result of the report changes (one printed interval of Theorem 6 is now rounded outwards), and no certificate data file changes. The complete list is in the section "Changes in version 1.2" of the report.
ER -