Repository navigation
Releases
v4.1.0: The Pólya–Vinogradov inequality with log log q for every character
Deep Bhattacharjee, Priyabrata Mandal and Ushashi Bhattacharya, The Pólya–Vinogradov inequality with log log q for every character. Version 4, single column (amsart), 53 pages.
M(χ) ≤ C √q log log q for every primitive Dirichlet character χ modulo q ≥ 3, with an absolute
constant C, where M(χ) is the largest modulus of a partial sum of χ, and |L'/L(1, χ)| ≤ C log log q
for every nonprincipal character (Theorem 1.5). The proof combines a transfer theorem for a single
character with the zero-free half-plane Re s > 11/12 for all Dirichlet L-functions; the half-plane
and its proof are due to OpenAI, and Appendix B gives the proof in full, with every external input
a published theorem.
New in this version:
- Section 8: the Euler–Kronecker constant γ_m of Q(ζ_m) satisfies γ_m ≤ 12 log m + 36 log log m + C
for every m ≥ 3, an order known before only under the generalised Riemann hypothesis (Ihara), and
γ_q ≥ −C log q log log q for primes q (Theorem 1.6). - No statement rests on an unrefereed source: the preprints of OpenAI are cited for attribution
only. - A shorter abstract and the standard structure (no note under the abstract); the new affiliation
of the first author. checks/python/eisenstein_check.pychecks the Gauss-sum and reciprocity identities of Lemma B.3
over the Eisenstein integers.
Version 3 (10.5281/zenodo.23672710) is hexagon:2610.00256v3; this version replaces it there.
Release v4.1.0 is the arXiv version: Priyabrata Mandal (Maulana Azad National Institute of
Technology Bhopal) joins as second author, and a title-page footnote names the Hexagon version
hexagon:2610.00256v3, https://hexagonmath.org/2610.00256v3, and the change of authors. The
mathematics is that of release v4.0.1 (10.5281/zenodo.23791644).
Files: the paper (PolyaVinogradovLogLog_final2.pdf), its LaTeX source with the figures
(PolyaVinogradovLogLog_final2-tex.zip) and the arXiv tarball
(PolyaVinogradovLogLog_final2-arxiv.tar.gz).
Contact: d.bhattacharjee@erl-forschung.de, itsdeep@live.com; for this repository egspl@proton.me.
v4.0.1: The Pólya–Vinogradov inequality with log log q for every character
Deep Bhattacharjee and Ushashi Bhattacharya, The Pólya–Vinogradov inequality with log log q for every character. Version 4, single column (amsart), 53 pages.
M(χ) ≤ C √q log log q for every primitive Dirichlet character χ modulo q ≥ 3, with an absolute
constant C, where M(χ) is the largest modulus of a partial sum of χ, and |L'/L(1, χ)| ≤ C log log q
for every nonprincipal character (Theorem 1.5). The proof combines a transfer theorem for a single
character with the zero-free half-plane Re s > 11/12 for all Dirichlet L-functions; the half-plane
and its proof are due to OpenAI, and Appendix B gives the proof in full, with every external input
a published theorem.
New in this version:
- Section 8: the Euler–Kronecker constant γ_m of Q(ζ_m) satisfies γ_m ≤ 12 log m + 36 log log m + C
for every m ≥ 3, an order known before only under the generalised Riemann hypothesis (Ihara), and
γ_q ≥ −C log q log log q for primes q (Theorem 1.6). - No statement rests on an unrefereed source: the preprints of OpenAI are cited for attribution
only. - A shorter abstract and the standard structure (no note under the abstract); the new affiliation
of the first author. checks/python/eisenstein_check.pychecks the Gauss-sum and reciprocity identities of Lemma B.3
over the Eisenstein integers.
Version 3 (10.5281/zenodo.23672710) is hexagon:2610.00256v3; this version replaces it there.
Release v4.0.1 is the journal version: it adds a title-page footnote naming the preprint
hexagon:2610.00256v3. The rest of the paper is that of release v4.0.0
(10.5281/zenodo.23789368).
Files: the paper (PolyaVinogradovLogLog_final2.pdf), its LaTeX source with the figures
(PolyaVinogradovLogLog_final2-tex.zip) and the arXiv tarball
(PolyaVinogradovLogLog_final2-arxiv.tar.gz).
Contact: d.bhattacharjee@erl-forschung.de, itsdeep@live.com; for this repository egspl@proton.me.
v4.0.0: The Pólya–Vinogradov inequality with log log q for every character
Deep Bhattacharjee and Ushashi Bhattacharya, The Pólya–Vinogradov inequality with log log q for every character. Version 4, single column (amsart), 53 pages.
M(χ) ≤ C √q log log q for every primitive Dirichlet character χ modulo q ≥ 3, with an absolute
constant C, where M(χ) is the largest modulus of a partial sum of χ, and |L'/L(1, χ)| ≤ C log log q
for every nonprincipal character (Theorem 1.5). The proof combines a transfer theorem for a single
character with the zero-free half-plane Re s > 11/12 for all Dirichlet L-functions; the half-plane
and its proof are due to OpenAI, and Appendix B gives the proof in full, with every external input
a published theorem.
New in this version:
- Section 8: the Euler–Kronecker constant γ_m of Q(ζ_m) satisfies γ_m ≤ 12 log m + 36 log log m + C
for every m ≥ 3, an order known before only under the generalised Riemann hypothesis (Ihara), and
γ_q ≥ −C log q log log q for primes q (Theorem 1.6). - No statement rests on an unrefereed source: the preprints of OpenAI are cited for attribution
only. - A shorter abstract and the standard structure (no note under the abstract); the new affiliation
of the first author. checks/python/eisenstein_check.pychecks the Gauss-sum and reciprocity identities of Lemma B.3
over the Eisenstein integers.
Version 3 (10.5281/zenodo.23672710) is hexagon:2610.00256v3; this version replaces it there.
Files: the paper (PolyaVinogradovLogLog_final2.pdf), its LaTeX source with the figures
(PolyaVinogradovLogLog_final2-tex.zip) and the arXiv tarball
(PolyaVinogradovLogLog_final2-arxiv.tar.gz).
Contact: d.bhattacharjee@erl-forschung.de, itsdeep@live.com; for this repository egspl@proton.me.
v3.0.0: The Pólya–Vinogradov inequality with log log q for every character
Deep Bhattacharjee and Ushashi Bhattacharya, The Pólya–Vinogradov inequality with log log q for every character. Final revision, single column (amsart), 51 pages.
M(χ) ≤ C √q log log q for every primitive Dirichlet character χ modulo q ≥ 3, with an absolute
constant C, where M(χ) is the largest modulus of a partial sum of χ, and |L'/L(1, χ)| ≤ C log log q
for every nonprincipal character (Theorem 1.5). From Chebyshev's bounds alone: for every ε > 0
at most q^ε of the primitive characters modulo q have M(χ) > C_ε √q log log q, and at most q^ε of
the nonprincipal characters have |L'/L(1, χ)| > C_ε log log q (Theorems 1.2 and 1.4). The proof
is a transfer theorem for a single character (a zero-free rectangle for the twists of χ by the
characters of small moduli gives the bound), proved by a contour shift, the circle method and a
weak Montgomery–Vaughan bound from Selberg's sieve; a count of the exceptional characters by a
zero-density theorem, proved in full in Appendix A with explicit constants; and the zero-free
half-plane Re s > 11/12 for all finite-order Hecke L-functions over Q(√−3), due to OpenAI, whose
proof Appendix B reproduces in full with credit to them, so that every statement of the paper is
proved in the text. OpenAI's Lean 4 proof of the wider half-plane Re s > 7/8 is compiled from
source and checked by the Lean kernel in this repository as an independent check.
This release adds Appendix B, retitles the paper, redraws the figures (the structure of the
proofs now includes the half-plane, and a new figure shows the parts of its proof) and sets the
paper in amsart. The results were first made public on 15 October 2026 as release v1.0.0 of this
repository; an earlier and incomplete version of the paper appeared on Hexagon as
hexagon:2610.00256.
Files: the paper (PolyaVinogradovLogLog_final2.pdf), its LaTeX source with the figures
(PolyaVinogradovLogLog_final2-tex.zip) and the arXiv tarball
(PolyaVinogradovLogLog_final2-arxiv.tar.gz).
Contact: d.bhattacharjee@erl-forschung.de, itsdeep@live.com; for this repository egspl@proton.me.
v2.0.0: The Pólya–Vinogradov inequality with log log q for all but q^ε characters
Deep Bhattacharjee and Ushashi Bhattacharya, The Pólya–Vinogradov inequality with log log q for all but q^ε characters. Final revision, single column, 26 pages.
For every ε > 0 there is a constant C_ε such that, for every modulus q ≥ 3, at most q^ε of
the primitive characters χ modulo q have M(χ) > C_ε √q log log q, where M(χ) is the largest
modulus of a partial sum of χ; in the same way at most q^ε of the nonprincipal characters have
|L'/L(1, χ)| > C_ε log log q. The proof is a transfer theorem for a single character (a
zero-free rectangle for the twists of χ by the characters of small moduli gives the bound),
proved by a contour shift, the circle method and a weak Montgomery–Vaughan bound from Selberg's
sieve, and a count of the exceptional characters by a zero-density theorem, which is proved in
full in Appendix A with explicit constants. Every statement is proved by hand from Chebyshev's
bounds, except Theorem 1.5, the bound for every character, which rests on the zero-free
half-plane Re s > 7/8 of OpenAI's preprint; that Lean 4 proof is compiled from source and
checked by the Lean kernel in this repository, with a Lean proof of Theorem 1.5 from it.
The results were first made public on 15 October 2026 as release v1.0.0 of this repository;
an earlier and incomplete version of the paper appeared on Hexagon as hexagon:2610.00256.
Files: the paper (PolyaVinogradovLogLog_final2.pdf), its LaTeX source with the figures
(PolyaVinogradovLogLog_final2-tex.zip) and the arXiv tarball
(PolyaVinogradovLogLog_final2-arxiv.tar.gz).
Contact: d.bhattacharjee@erl-forschung.de, itsdeep@live.com; for this repository egspl@proton.me.
v1.6.0 - submission and verification release
Submission and verification revision v1.6.0
This consolidated submission-package release retains the unchanged article and credited zero-free theorem, the matching complete formal-source ZIP, historical axiom-audit receipts and scoped independent-input research notes. No new mathematical theorem or independent zero-free replacement is claimed by this packaging revision.
The formal source tree is unchanged from the audited 19 original, seven closed endpoint and 20 analytic component reports. This release verifies that source identity and archive bytes. It does not claim a newly rerun full Lean build or a proof of the proposed independent mean-square input.
The manuscript PDF and source ZIP retain their previously checked bytes.
Existing bylines and source credits are preserved. The package includes
Hexagon fields and journal preparation files for Journal of Number Theory.
New material awaits author review. No journal or Hexagon submission,
acceptance, full Lehmer closure or independent zero-free proof is claimed.
Previous package DOI: https://doi.org/10.5281/zenodo.23637263
The new archival DOI is confirmed below; archived payload checks passed.
Package DOI: https://doi.org/10.5281/zenodo.23638793 .
Complete package folder: https://github.com/creelie/Breakthrough-problems-3/tree/v1.6.0/Paper_6%20package .
All package payloads were verified against the archived release commit.
Complete submission package ZIP: Paper_6_v1.6.0_submission_package.zip.
This ZIP was assembled after DOI verification and includes the post-publication DOI receipt; the immutable Zenodo source archive remains separately identified.
v1.5.0 - research construction and reproducibility revision
This is a research and reproducibility revision. No independent zero-free replacement is proved.
The 29-page article is unchanged and continues to credit OpenAI's zero-free theorem. The package adds a separate complete nonstandard formal-source ZIP with licences and pins, source/import manifests, rechecked historical reports and build instructions. The unchanged source tree has the completed 19 original, seven endpoint and 20 analytic component axiom reports; no new full Lean build is claimed for this documentation revision.
The research notes prove the conditional extraction calculation at H=D^(21/20), giving a sufficient 15/16 boundary if its specialized mean-square input were established. That input remains unproved. A coefficient-independent shortcut is obstructed by sixth-power rows. The classical Gauss/Jacobi identity and CRT construction are derived with finite sums; all 21 prime-field polynomial controls pass exactly. These arithmetic identities do not prove the needed off-diagonal estimate and are not claimed as new discoveries.
Paper_6 package includes the unchanged article PDF/TeX, separate research source materials, provenance, portal fields and unsent Journal of Number Theory correspondence. Deep Bhattacharjee and Ushashi Bhattacharya remain coauthors. Added research notes await author review. The source bundle is separate from the Hexagon manuscript ZIP; no new manuscript version or submission is claimed.
Existing preprint: https://hexagonmath.org/2610.00256v2
Prior package DOI: https://doi.org/10.5281/zenodo.23629649
A new DOI archives this scope; it does not certify a new independent theorem, peer review or journal acceptance.
Package DOI: https://doi.org/10.5281/zenodo.23637263 .
Complete package folder: https://github.com/creelie/Breakthrough-problems-3/tree/v1.5.0/Paper_6%20package .
All package payloads were verified against the archived release commit.
v1.4.0 — Hexagon and journal submission package
The optimal log log q bound in the Pólya-Vinogradov inequality — Deep Bhattacharjee and Ushashi Bhattacharya.
This release adds Paper_6 package, a reviewed upload package for Hexagon/Overleaf and Journal of Number Theory. It contains the manuscript PDF, minimal compilable TeX ZIP, AI provenance, complete Hexagon submission fields, journal details, compiled unsent cover letter with editable source, and an author-prepared AI declaration. PDFs have no row or margin line numbering.
Validation: each extracted article ZIP compiled with pdfLaTeX and matched the existing manuscript's complete extracted text and page count. Supporting PDFs passed compilation and visual review; ZIP integrity and per-file SHA-256 manifests passed. This packaging does not create a new mathematical result.
Proof/check scope: The complete current source build, all 19 original and seven closed endpoint axiom reports, and all 20 analytic component reports passed. The reports use only propext, Classical.choice and Quot.sound. The separate written cyclotomic identification is not a new Lean statement, and the revised exposition is not a line-by-line formalisation. No independent zero-free proof is claimed.
The manuscript derives the optimal-order log log q character-sum bound and related logarithmic-derivative results from the credited zero-free theorem, with explicit parameters and a smooth Mobius equivalence. The complete formal source and exact endpoint audit passed. OpenAI's zero-free proof remains credited; no newly independent proof is claimed.
The existing byline, licences and AI attribution are retained. The submitting author explicitly declared mixed human/AI text and human checking and understanding of all parts. The standalone declaration is author-prepared and is not an official signed publisher form. No journal submission, acceptance, new arXiv identifier or later Hexagon identifier is claimed.
Earlier preprint: https://hexagonmath.org/2610.00256v2 . Earlier manuscript/source DOI: https://doi.org/10.5281/zenodo.23629139 . The new package archive is published and its contents have been checksum-verified.
Package DOI: https://doi.org/10.5281/zenodo.23629649 .
Complete package folder: https://github.com/creelie/Breakthrough-problems-3/tree/v1.4.0/Paper_6%20package .
All package payloads were verified against the archived release commit.
v1.3.0 — Source-complete research revision
Source-complete research revision of The optimal log log q bound in the Pólya–Vinogradov inequality, by Deep Bhattacharjee and Ushashi Bhattacharya.
The current article is one 29-page aomart manuscript without margin line numbers or an appended third-party article. The archive supplies all 2,997 nonstandard Lean source files, including OpenAI's credited 7/8 nonvanishing proof, with the pinned Lean/Mathlib manifest, source provenance, original notices, verification logs and rebuild instructions. The new Möbius modules implement the arbitrary-boundary equivalence, reciprocal estimate, Mellin argument and finite-sum identity. Standard library downloads remain necessary on a clean installation.
Verification at publication: the original complete source build and all 19 original theorem axiom reports passed. The original theorem sources are unchanged; its historical revised-wrapper failure is explicitly preserved in the receipt. The new Linux analytic component source build and all 20 named axiom reports passed. Only propext, Classical.choice and Quot.sound occur in those reports. The fresh current root build and the new closed endpoint audit are still pending and are not reported as passed. See PROOF-COVERAGE.md and the receipts for the exact scopes.
The PDF passed local and desktop compilation, citation checks and visual review. A read-only GitHub source-export check confirmed one current article, no historical compiled distributions, and matching contents of every bundled Lean source. Historical assets remain available in Git and earlier releases.
This release retains the mathematical contribution and authorship of the supplied OpenAI zero-free proof. No newly independent zero-free proof by the manuscript authors is claimed. It is an archival research revision, not a certificate that every line of the written exposition is formally verified or that a journal has accepted the paper. The cover letter is unsent; no journal submission is performed.
Earlier public preprint: https://hexagonmath.org/2610.00256v2 . The current PDF and notes are in journal/paper6-annals-review/. Zenodo publication and the archived source checksums were confirmed before creating the final submission ZIP. The DOI and ZIP links are below.
Proof/source archive DOI: https://doi.org/10.5281/zenodo.23629139 .
Final submission ZIP: https://github.com/creelie/Breakthrough-problems-3/releases/download/v1.3.0/Paper_6_Polya_Vinogradov_v1.3.0_research_submission.zip .
The DOI was confirmed before creating the final ZIP.
The Pólya–Vinogradov inequality with log log q — aomart research revision
The Pólya–Vinogradov inequality with log log q
The paper retains the external 7/8 zero-free theorem. Its independent self-contained proof has not been supplied here.
This aomart research revision is not a proof of all eight requested closures. No journal acceptance or new human review is claimed. The package has been compiled and visually checked; all eight PDFs have no unresolved references, overfull boxes or off-page text in the recorded check. The archival record for this exact release version has been confirmed.
Files committed with this release target:
- 01-PolyaVinogradovLogLog-aomart.pdf
- 01-PolyaVinogradovLogLog-aomart-source.zip
- 01-PolyaVinogradovLogLog-scope.md
The repository has an active Zenodo release webhook. No guessed or inherited DOI is assigned to this revision.
Verified Zenodo version DOI: 10.5281/zenodo.23601915. The DOI identifies this archived research revision; the scientific scope stated above is retained.