The Riemann Hypothesis

The Riemann Hypothesis

OpenAI's claim

The two selected manuscripts claim zero-free half-planes for the Riemann zeta function, together with broader Dirichlet and finite-order Hecke families. Manuscript 1 gives the boundary $11/12$, and manuscript 2 gives $7/8$. Neither boundary is the critical line $1/2$ required by this problem. Manuscript numbers follow the order in Sources.

Mathematical statement of the claimed result

Manuscript 1.

For $K=\mathbb Q(\sqrt{-3})$ and every finite-order Hecke character $\chi$ over $K$, the claimed zero-free assertion, with meromorphic poles excluded from evaluation, is \[ \operatorname{Re}s>11/12\quad\Longrightarrow\quad L_K(s,\chi)\ne0. \] It includes every Dirichlet $L$-function and the Riemann zeta function. The accompanying prime-progression claim uses $\pi(x;q,a)$ and asserts, with an absolute effective constant, \[ \sup_{\substack{1\le q\le x\$a,q)=1}} \left|\pi(x;q,a)-\frac1{\varphi(q)}\int_2^x\frac{dt}{\log t}\right| \ll x^{11/12}\log x\quad(x\ge2). \] The critical-line requirement $\operatorname{Re}\rho=1/2$ is not a consequence of this half-plane claim.

Manuscript 2.

For every finite-order Hecke character of $F=\mathbb Q(\sqrt{-3})$, every Dirichlet character and the zeta function, the principal claim is \[ \operatorname{Re}s>7/8\quad\Longrightarrow\quad L(s)\ne0, \] where a principal character's pole at $s=1$ is allowed. The weaker displayed $11/12$ theorem follows within this scope. The cited arithmetic consequence gives absolute $A,C>0$ such that every odd prime $p$ has least positive quadratic nonresidue \[ n(p)\le C(\log p)^A. \] Thus $n(p)\ll_{\delta} p^{\delta}$ for each fixed $\delta>0$. Given binary $p$ and $a\in\mathbb F_p$, a deterministic algorithm of time polynomial in $\log p$ is further claimed to return a square root of $a$ or correctly declare it nonexistent. The half-plane assertion leaves zeros between the critical line and $7/8$ untreated.

Outline of the argument

For manuscript 1 (family 003; 2026-10-05), twisted Möbius sums are enlarged to a family before Poisson summation. A recursive estimate for the dual mean square, followed by inversion of cube factors, supplies the saving used to exclude a reciprocal-function singularity.

For manuscript 2 (family 003; 2026-09-30), smoothed character sums are estimated in two ways: directly by large-sieve and recursive moment bounds, and through a Mellin transform involving the reciprocal of an $L$-function. Poisson summation and sextic residue characters align the two expressions so that their savings force continuation of the reciprocal.

Formal verification

For manuscript 2 (family 003; 2026-09-30), OpenAI lists Lean statements for the $7/8$ zero-free claims for zeta, Dirichlet functions and finite-order Hecke functions over $\mathbb Q(\sqrt{-3})$. This covers the asserted half-plane statements and does not establish the critical-line conclusion asked here. No Lean formalization is listed at this snapshot for manuscript 1 (family 003; 2026-10-05).

The selected Comparator checks are DirichletSevenEighths.lean, HeckeSevenEighths.lean, QuasiRiemannHypothesis.lean.

Their named declarations include OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.SevenEighths.HeckeFamily.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

• OpenAI, The Quasi-Riemann Hypothesis, 2026-10-05; citation key OAI:The-Quasi-Riemann-Hypothesis-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re (s)>7/8, 2026-09-30; citation key OAI:The-Quasi-Riemann-Hypothesis-September-30-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 003: Lean scope.
• Comparator DirichletSevenEighths.json.
• Comparator DirichletSevenEighths.lean.
• Comparator HeckeSevenEighths.json.
• Comparator HeckeSevenEighths.lean.
• Comparator QuasiRiemannHypothesis.json.
• Comparator QuasiRiemannHypothesis.lean.
• OpenAI catalogue entry.

The Riemann Hypothesis

OpenAI's claim

The two selected manuscripts claim zero-free half-planes for the Riemann zeta function, together with broader Dirichlet and finite-order Hecke families. Manuscript 1 gives the boundary $11/12$, and manuscript 2 gives $7/8$. Neither boundary is the critical line $1/2$ required by this problem. Manuscript numbers follow the order in Sources.

Mathematical statement of the claimed result

Manuscript 1.

For $K=\mathbb Q(\sqrt{-3})$ and every finite-order Hecke character $\chi$ over $K$, the claimed zero-free assertion, with meromorphic poles excluded from evaluation, is \[ \operatorname{Re}s>11/12\quad\Longrightarrow\quad L_K(s,\chi)\ne0. \] It includes every Dirichlet $L$-function and the Riemann zeta function. The accompanying prime-progression claim uses $\pi(x;q,a)$ and asserts, with an absolute effective constant, \[ \sup_{\substack{1\le q\le x\$a,q)=1}} \left|\pi(x;q,a)-\frac1{\varphi(q)}\int_2^x\frac{dt}{\log t}\right| \ll x^{11/12}\log x\quad(x\ge2). \] The critical-line requirement $\operatorname{Re}\rho=1/2$ is not a consequence of this half-plane claim.

Manuscript 2.

For every finite-order Hecke character of $F=\mathbb Q(\sqrt{-3})$, every Dirichlet character and the zeta function, the principal claim is \[ \operatorname{Re}s>7/8\quad\Longrightarrow\quad L(s)\ne0, \] where a principal character's pole at $s=1$ is allowed. The weaker displayed $11/12$ theorem follows within this scope. The cited arithmetic consequence gives absolute $A,C>0$ such that every odd prime $p$ has least positive quadratic nonresidue \[ n(p)\le C(\log p)^A. \] Thus $n(p)\ll_{\delta} p^{\delta}$ for each fixed $\delta>0$. Given binary $p$ and $a\in\mathbb F_p$, a deterministic algorithm of time polynomial in $\log p$ is further claimed to return a square root of $a$ or correctly declare it nonexistent. The half-plane assertion leaves zeros between the critical line and $7/8$ untreated.

Outline of the argument

For manuscript 1 (family 003; 2026-10-05), twisted Möbius sums are enlarged to a family before Poisson summation. A recursive estimate for the dual mean square, followed by inversion of cube factors, supplies the saving used to exclude a reciprocal-function singularity.

For manuscript 2 (family 003; 2026-09-30), smoothed character sums are estimated in two ways: directly by large-sieve and recursive moment bounds, and through a Mellin transform involving the reciprocal of an $L$-function. Poisson summation and sextic residue characters align the two expressions so that their savings force continuation of the reciprocal.

Formal verification

For manuscript 2 (family 003; 2026-09-30), OpenAI lists Lean statements for the $7/8$ zero-free claims for zeta, Dirichlet functions and finite-order Hecke functions over $\mathbb Q(\sqrt{-3})$. This covers the asserted half-plane statements and does not establish the critical-line conclusion asked here. No Lean formalization is listed at this snapshot for manuscript 1 (family 003; 2026-10-05).

The selected Comparator checks are DirichletSevenEighths.lean, HeckeSevenEighths.lean, QuasiRiemannHypothesis.lean.

Their named declarations include OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.SevenEighths.HeckeFamily.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

• OpenAI, The Quasi-Riemann Hypothesis, 2026-10-05; citation key OAI:The-Quasi-Riemann-Hypothesis-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re (s)>7/8, 2026-09-30; citation key OAI:The-Quasi-Riemann-Hypothesis-September-30-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 003: Lean scope.
• Comparator DirichletSevenEighths.json.
• Comparator DirichletSevenEighths.lean.
• Comparator HeckeSevenEighths.json.
• Comparator HeckeSevenEighths.lean.
• Comparator QuasiRiemannHypothesis.json.
• Comparator QuasiRiemannHypothesis.lean.
• OpenAI catalogue entry.

The Riemann Hypothesis

Definitions and mathematical statement

For $s\in{\mathbb{C}}$ with $\Re s>1$, define \[ \zeta(s)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p\text{ prime}}(1-p^{-s})^{-1}. \] Use its meromorphic continuation to ${\mathbb{C}}$, whose only pole is at $s=1$. The assertion is \[ \forall\rho\in{\mathbb{C}},\qquad \left(0<\Re\rho<1\ \text{and}\ \zeta(\rho)=0\right) \ \Longrightarrow\ \Re\rho=\tfrac12. \] The open critical strip excludes the negative even integer zeros. The assertion concerns locations, not multiplicities, of zeros.

Formulation and concept references: [S1].

Short English statement

Are all zeta zeros between the vertical lines zero and one located on the halfway line?

Sources

• S1 Enrico Bombieri, Problems of the Millennium: the Riemann Hypothesis, Section I and the prime-counting equivalence. https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf. Catalogue formulation source. Consulted 24 September 2026: Section I, definition and zero-location statement.
• S2 Riemann Hypothesis. https://www.claymath.org/millennium/riemann-hypothesis/. Catalogue research/status reference; not a proof endorsement.
• S3 A new proof that more than 2/3 of the zeros of the Riemann zeta function are simple and on the critical line. https://arxiv.org/abs/2609.02882. Catalogue research/status reference; not a proof endorsement.
• S4 Lamzouri lecture at Max Planck Institute for Mathematics, September 3, 2026. https://math-events.uni-bonn.de/event/1560/. Catalogue research/status reference; not a proof endorsement.
• S5 On the Extended, Generalized, and Grand Riemann Hypotheses: A Unified Approach Based on the General Properties of L-Functions — Weicun Zhang, v19. https://www.preprints.org/manuscript/202506.0481. Catalogue research/status reference; not a proof endorsement.

• E1 Dave Platt and Tim Trudgian, The Riemann hypothesis is true up to $3\cdot10^{12}$, arXiv:2004.09765v1 (21 April 2020), Theorem 1 and Section 2; published in Bulletin of the London Mathematical Society $2021$. https://arxiv.org/pdf/2004.09765v1. Research reference: rigorous bounded-height verification, including completeness of the zero count. Consulted 24 September 2026.

• E2 Larry Guth and James Maynard, New large value estimates for Dirichlet polynomials, Annals of Mathematics 203 $2026$, no. 2, 623--675, DOI: 10.4007/annals.2026.203.2.6, Theorem 1.2. https://annals.math.princeton.edu/2026/203-2/p06. Author manuscript: arXiv:2405.20552v2 (7 April 2026), https://arxiv.org/pdf/2405.20552v2. Research reference: zero-density estimate; publication metadata and the stated theorem were checked. Consulted 24 September 2026.

• E3 Michael Griffin, Ken Ono, Larry Rolen and Don Zagier, Jensen polynomials for the Riemann zeta function and other sequences, arXiv:1902.07321v2 (31 March 2019), Section 1, Theorems 1--2; published in Proceedings of the National Academy of Sciences $2019$. https://arxiv.org/pdf/1902.07321v2. Research reference: the fixed-degree, sufficiently-large-shift quantifiers, not all-degree hyperbolicity. Consulted 24 September 2026.

• E4 Brad Rodgers and Terence Tao, The de Bruijn--Newman constant is non-negative, Forum of Mathematics, Pi 8 $2020$, e6, DOI: 10.1017/fmp.2020.6. Author manuscript arXiv:1801.05914v5 (3 July 2021), Section 1, equations $1$--$4$, and the main theorem. https://arxiv.org/pdf/1801.05914v5. Research reference: the theta-kernel normalization and the endpoint obstruction. Consulted 24 September 2026.

• E5 David W. Farmer, Jensen polynomials are not a plausible route to proving the Riemann Hypothesis, arXiv:2008.07206, Sections 2--4. https://arxiv.org/pdf/2008.07206. Research reference: differentiation and information loss; the paper's methodological assessment is not used as an impossibility theorem. Consulted 24 September 2026.

• E6 Jonathan Holland, A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $\xi$-function, arXiv:2608.08682v1 (9 August 2026), Theorem 1.1. https://arxiv.org/html/2608.08682v1. Research reference: recent manuscript, statement inspected, full proof not independently audited here; not used as a premise of any lemma. Consulted 24 September 2026.

• E7 Kiran S. Kedlaya, More on the zeroes of $\zeta$, MIT 18.785: Analytic Number Theory, spring 2007, Section 1, Lemma 1 and Theorems 2--3. https://kskedlaya.org/18.785/zeroes.pdf. Research reference: growth of the completed function, zero counting, and Hadamard factorization. Consulted 24 September 2026.

• E8 Paul Garrett, 12. Weierstrass and Hadamard products, author lecture notes, 3 February 2021, Section 3 (Hadamard products). https://www-users.cse.umn.edu/~garrett/m/complex/notes_2020-21/12_Hadamard_products.pdf. Research reference: canonical products and the finite-order factorization theorem. Consulted 24 September 2026.

• E9 Youness Lamzouri, A new proof that more than $2/3$ of the zeros of the Riemann zeta function are simple and on the critical line, arXiv:2609.02882v2 (8 September 2026), Theorem 1.1, Proposition 2.1 and Sections 2--3. https://arxiv.org/html/2609.02882v2. Version-specific research supplement to [S3]: the stated bound and method were inspected. Neither a complete analytic proof audit nor a formal-certificate audit was performed. Consulted 24 September 2026.