Openai 2026 quasi riemann hypothesis zero free half plane 11 12

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corollary_1_2: The prime number theorem in arithmetic progressions with an absolute effective error term x^{11/12} log x, uniform over all moduli q ≤ x, stated as a consequence of Theorem 1.1 by the explicit formula and not proved in the manuscript, checked at claims level only and unverified here.

theorem_1_1: The manuscript's main claim, checked at claims level only and unverified here: no finite-order Hecke L-function over Q(sqrt(-3)), hence no Dirichlet L-function and not the zeta function, has a zero with real part above 11/12, proved from a mean-square bound for sextic-twisted Möbius sums.

OpenAI, The Quasi-Riemann Hypothesis, OpenAI Math Release preprint, October 5, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/The-Quasi-Riemann-Hypothesis-October-5-2026; the held PDF, paper2.pdf in the release, is retained as openai_2026_quasi_riemann_hypothesis_zero_free_half_plane_11_12.pdf, and the release's TeX bundle in that folder is the TeX source cited on this card.

@misc{OAI:The-Quasi-Riemann-Hypothesis-October-5-2026,
  author = {{OpenAI}},
  title = {{The Quasi-Riemann Hypothesis}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf}{OAI:The-Quasi-Riemann-Hypothesis-October-5-2026}},
  year = {2026}
}

The release's root README states that its manuscripts and proof artifacts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that "Not all have accompanying Lean formalizations" and that "Some of the unformalized results could have issues". Its account of the method says the results were obtained by one fixed procedure, then adds: "Exceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function", which refers to the family's zero-free-region work without naming a manuscript, and "the writeup for the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability", which names this 11/12 writeup. The manuscript's own README gives the author as OpenAI and the date as October 5, 2026 and adds "This paper was written with human assistance."; the text itself names OpenAI as sole author and prints no personal names, no acknowledgments and no further statement on how it was produced. These are the source's statements about its own provenance, recorded here as attestations and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

The release's Lean catalog (lean/formalization.yaml) does not name this manuscript; the family's formalization is recorded on the companion card The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8.

The release files this manuscript in one family with two others, both held in this library: The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8 (September 30, 2026), whose main theorem the present text cites as the "stronger zero-free region" (p. 2) of which its own Theorem 1.1 is "a natural intermediate step" (p. 2), so this manuscript is the alternate proof with the weaker exponent; and Uniform exclusion of Landau–Siegel zeros (October 1, 2026), a companion on the real-zero consequence that the present abstract states in one sentence.

Read status: claims checked for Theorem 1.1, Corollary 1.2 and the statements of Propositions 3.1, 4.5, 5.1, 5.2 and 5.4 and Lemma 5.3, read clause by clause in the release's TeX source (paper2.tex, labels thm:main, cor:primes-ap, thm:ms, prop:poisson-reduction, prop:canonical, prop:R, lem:cube-reduction, prop:transfer) on 2026-10-07; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed. Corollary 1.2 has no proof in the manuscript (it is referred to a "standard explicit-formula argument" (p. 2) in Davenport), and the consequences listed after it in Section 1 are stated with citations and not proved there.

Contents

The PDF has 49 pages. Theorem and equation numbers below are the PDF's; the TeX labels are given where a page cites them.

• Section 1, Introduction (pp. 1--3). Defines $L(s,\chi)$ for a primitive character of conductor $q$, recalls the Generalized Riemann Hypothesis and the name "quasi-Riemann Hypothesis" (p. 1) for a zero-free half-plane $\operatorname{Re}s>1-\varepsilon$ (cited to Bettin--Gonek, Murty--Sankaranarayanan and Bhowmik--Ruzsa), and asks the same with one $\varepsilon$ for every primitive Dirichlet character. States Theorem 1.1 (every finite-order Hecke $L$-function over $K=\mathbb Q(\sqrt{-3})$, hence every Dirichlet $L$-function and $\zeta(s)$, has no zeros for $\operatorname{Re}s>11/12$), says that Section 3 proves it from Proposition 3.1 whose proof ends in Section 5.6, and calls it an intermediate step toward the companion's $7/8$. States Corollary 1.2 (primes in progressions with error $\ll x^{11/12}\log x$ uniformly for $q\le x$) as following "by a standard explicit-formula argument" (p. 2) from Davenport's Chapters 19--20; no proof is given. A paragraph then lists consequences of Theorem 1.1, each with a citation and none proved in the text: a bound of the form $(\log p)^{C}$ for the least quadratic nonresidue modulo an odd prime $p$ (Rodosskii's method as in Montgomery--Vaughan, Theorem 13.12; also Bhargava, Ivanyos, Mittal and Saxena, Theorem 6.7), which proves Vinogradov's conjecture; deterministic polynomial-time square roots modulo $p$ by Tonelli--Shanks; a deterministic polynomial-time form of Miller's primality test (noting that AKS already gives one); the effective class-number bound $h(D)\gg\sqrt{|D|}/\log\log|D|$ for negative fundamental discriminants by Littlewood's short Euler-product argument; and the completeness of Euler's list of 65 idoneal numbers through Elsenhans, Klüners and Nicolae's Theorem 2, Weinberger, and Grube's reduction as presented by Kani. Subsection 1.1, Prior work: Hadamard and de la Vallée Poussin, the Grönwall--Titchmarsh region (1.1) with its one possible real exception, the Landau--Siegel zero, which the text says Theorem 1.1 "rules out" (p. 2); the Vinogradov--Korobov region (1.2) for $\zeta$; and the method's lineage in Kubota's and Patterson's cubic theta function, Heath-Brown and Patterson, Heath-Brown's Kummer paper, Dunn and Radziwiłł, and Heath-Brown's quadratic large sieve. Subsection 1.2 gives the organization; 1.3 the notation: $\mathcal O=\mathbb Z[\omega]$, the norm $\mathrm N_{K/\mathbb Q}$, dyadic ranges, the size parameter $D\ge2$, and $A\preccurlyeq B$ for $A\ll_\varepsilon D^\varepsilon B$ for every $\varepsilon>0$.
• Section 2, Outline of the argument (pp. 3--11). A sketch "suppressing various coprimality conditions, local factors, and details of smoothing" (p. 3). Step 1: for a finite-order Hecke character $\nu$ of $K$, a power saving $A_1(D)\ll D^{1-\delta+\varepsilon}$ for $A_1(D)$, the Möbius sum over ideals of norm about $D$ twisted by $\nu$ and smoothed by a weight $W$, gives the zero-free half-plane (the Hecke analogue of Littlewood's Möbius criterion). Step 2: embed $A_1(D)$ in the family $A_u(D)$ twisted by the sextic residue symbol $\chi_n(u)=(u/n)_6$ of the primary generator $n$; the target (2.3) is a mean square $\ll D^{1+\varepsilon}H$ over $0<\mathrm N(u)\le H=D^{1+\vartheta}$, $0<\vartheta\le1/10$; since $\chi_n(p^6)$ is the indicator of $p\nmid n$, the rows $u=p^6$ for primes of norm about $H^{1/6}$ recover $A_1(D)$ up to $O(D/Y)$, and Landau's prime ideal theorem gives enough of them for $A_1(D)\ll D^{11/12+5\vartheta/12+\varepsilon}$. Step 3: Poisson summation in $u$ produces sextic Gauss sums; the identity $\mu(n)\gamma_{-1}(n)=\chi_n(-1)G(n)^{-1}\overline{\alpha(n)}\gamma_2(n)$ (after Hasse and Heath-Brown) turns the Möbius coefficient into the cubic Gauss-sum coefficient $\overline{\alpha(n)}\gamma_2(n)$ and the problem into the dual mean square (2.9) of the column sums $B_h(X)$. Step 4: Patterson's formula $c_\theta(nb^3)=3^{5/2}|b|\,\overline{\chi_n(\lambda)^2}\gamma_2(n)$ identifies these coefficients with Fourier coefficients of Kubota's cubic theta function (Dunn--Radziwiłł's normalization); the sum is completed by cube indices $nb^3$ to $T_h(X)$; a variant of Dunn--Radziwiłł's theta transformation sends $T_h(X)$ to dual sums over three cusps whose twist at each prime $p\mid h$ is the quadratic character $\chi_p^3$ (2.14), so the Goldmakher--Louvel quadratic large sieve gives the completed bound (2.15) with the factor $\mathcal H+\mathcal H^2/X$ and avoids the $(ML)^{2/3}$ term of the general higher-order large sieve. Step 5: Möbius inversion in the cube variable (2.18) returns to the uncompleted sum; divisors up to a cutoff $H_c$ (2.21) are handled by the completed bound, the larger ones by a transfer estimate obtained from two further Poisson summations with an enlarged row range (after Goldmakher--Louvel and Heath-Brown), which expresses the remaining mean square through mean squares of the same type at smaller scales with the row range contracted by $D^{-2\vartheta}$ per step; after $O_\vartheta(1)$ steps counting finishes, in the admissible-exponent manner of Heath-Brown's quadratic large sieve. The sketch closes by naming the exact family $\mathcal E(\mathcal H,X,F)$ with auxiliary twists $\chi_n(f)^4$.
• Section 3, From the mean-square estimate to the zero-free region (pp. 11--12). Fixes $\nu$ and a finite set $S$ of prime ideals containing those above $2$, $3$ and the conductor of $\nu$. Proposition 3.1 (label thm:ms): for fixed $0<\vartheta\le1/10$ and $\varepsilon>0$ there is $k=k(\vartheta,\varepsilon)$ with $\sum_{0<\mathrm N(u)\le D^{1+\vartheta}}|A_u(D)|^2 \ll_{\nu,S,I,\vartheta,\varepsilon} (\max_{j\le k}|W^{(j)}|_\infty)^2D^{2+\vartheta+\varepsilon}$ for every smooth $W$ supported in a compact $I\subset(0,\infty)$ and $D\ge2$. Proof of Theorem 1.1 from it: the prime-sixth-power extraction (3.2) gives $A_1(D)\ll D^{11/12+\varepsilon}$ (3.3); the Mellin transform $\mathcal M_W(s)=\int_0^\infty A_1(D)D^{-s}\,dD/D$ is holomorphic on $\operatorname{Re}s>11/12$ and equals $\widehat W(s)/L_K^S(s,\nu)$ there by Hecke's continuation and the identity theorem, so a zero $\varrho$ with the choice $W(y)=y^{-\varrho}\phi(y)$ gives $0=\widehat W(\varrho)>0$; the Dirichlet case follows from $L_K(s,\chi\circ\mathrm N)=L(s,\chi)L(s,\chi\chi_{-3})$ up to Euler factors nonzero for $\operatorname{Re}s>0$, with $L(1,\chi_{-3})>0$ excluding a pole--zero cancellation at $s=1$.
• Section 4, Poisson summation and the dual mean square (pp. 12--18). Lemma 4.1 (label lem:arithmetic) collects the Gauss--Jacobi, reciprocity and ray-class identities: a unit-modulus function $G$ and a symmetric $\{\pm1\}$-valued bicharacter $\mathcal R$ on a fixed ray class group with $\chi_b(a)=\mathcal R(a,b)\chi_a(b)$, $\gamma_2(n)^3=\mu(n)\alpha(n)$, $\gamma_1\gamma_2=\mu\alpha G$, $G(n)=\overline{\chi_n(4)}\gamma_3(n)$, $\gamma_1\gamma_{-1}=\chi_n(-1)$, and the consequences (4.4)--(4.7). Lemma 4.2: lattice Poisson summation for a primitive character with an excluded ideal (4.8). Definition 4.3: the dual mean square $\mathcal E(\mathcal H,X,F;\xi,W)$ (4.9) over squarefree auxiliary twists $f$ of norm in $[F,2F)$ and rows $0<\mathrm N(k)\le\mathcal H$. Lemma 4.4: removal of an exclusion ideal at the cost of a divisor factor, preserving $XF$. Proposition 4.5 (label prop:poisson-reduction): if (4.12), $\mathcal E\ll\|W\|_{C^J(I)}^2D^\varepsilon XF$, holds in the ranges (4.11) ($X=D/(BF)$, $\mathcal H\le CD^2/(HB^2)$), then Proposition 3.1 holds; the proof expands the square, extracts the common factor of the two columns, applies Lemma 4.2, converts the paired Gauss sums with (4.5)--(4.7), changes variables to coprime squarefree $b,f$ and a free row $k$ (4.16), partitions dyadically, and separates the coupled weight with Lemma B.2.
• Section 5, Iteration of the dual mean-square estimate (pp. 18--22). Proposition 5.1 (label prop:canonical): for fixed $\kappa>0$, $C_0\ge1$, under $\mathcal H,X,F\ge1$, $\Sigma=XF\le D^{C_0}$ and $\mathcal H\le\Sigma D^{-\kappa}$, one has $\mathcal E\ll\|W\|_{C^J(I)}^2D^\varepsilon\Sigma$. Subsection 5.1 defines the completed sum $T(X;\Psi)$ (5.3) with cube index $b$ and the twist $\Psi_k(n)=\xi(n)\chi_n(k)\chi_n(f)^4$ (5.4). Proposition 5.2 (label prop:R): $\sum_{0<\mathrm N(k)\ll\mathcal H}|T(X;k,f)|^2 \ll D^\varepsilon|W|{C^J(I)}^2(\mathcal H+\mathcal H^2\mathrm N(f)/X)$ for $1\le\mathcal H,X,\mathrm N(f)\le D^{C_0}$. Lemma 5.3 (label lem:cube-reduction): with $H_c^3=\min(X,X^2/\mathcal H^2)$ and $L_b=X/\mathrm N(b)^3$, $\mathcal E(\mathcal H,X,F)\ll D^\varepsilon(\Sigma|W|^2 +\sup{\mathrm N(b)>H_c,,L_b>1}\mathcal E(\mathcal H,L_b,F))$, by Möbius inversion (5.8) and a short/long split. Proposition 5.4 (label prop:transfer): the smoothed mean square $\mathcal A(W)$ at column scale $L$ is $\ll D^\varepsilon\Sigma\|W\|_{C^{4m+12}(I)}^2(1+\sup\mathcal E'/\Sigma')$ over mean squares with $\mathcal H'\le\mathcal HL/(\Sigma F)$, $\mathcal H'/\Sigma'\le\mathcal H/\Sigma$, $\Sigma'\le L$. Subsection 5.5 proves Proposition 5.1 by induction on $j$ with $\mathcal H\le D^{j\kappa}$, the base case by counting and the step by Lemma 5.3 and Proposition 5.4 (each step contracts the row range by $D^{-2\kappa}$). Subsection 5.6 proves Proposition 3.1 with $\kappa=\vartheta/2$, $C_0=2$; the heading names "the exponent 11/12" (p. 22).
• Section 6, Proof of the completed mean-square estimate (pp. 22--29). Lemma 6.1 realizes $T(X;k,f)$ as a twisted sum of the squarefree-and-cube coefficients of $\overline\theta$ (6.3) and the twisted theta function as a finite sum of translates (6.4). Subsection 6.2 defines the local factors $B_{p,j}$ (6.5), the transformed weight $V_*^\sharp$ (6.6) through a gamma quotient, the three cusp representatives (6.7) and cusp coefficients $d_0,d_+,d_-$ (6.8). Proposition 6.2 (label lem:reflection): $T(X;\Psi)$ is a sum of $O(2^{|\mathcal P|})$ dual sums (6.9) over $\ell\in\lambda^{-4}\mathcal O$ with weight $V_*^\sharp(\mathrm N(\ell)X/\mathrm N(c)^2)$. Lemma 6.3: uniformity of the cusp data in the twist within fixed ray classes. Lemma 6.4: support $\ell=u\lambda^mnb^3$, the bound $|d(\ell)|\le27\cdot3^{m/6}|b|$ and rapid decay of $V_*^\sharp$. Lemma 6.5: the quadratic large sieve over $K$ (Goldmakher--Louvel, Theorem 1.1), with the factor $(\mathcal HU)^\varepsilon(\mathcal H+U)$. Lemma 6.6: the completed bound for squarefree rows, by transforming, reindexing at the active primes and applying Lemma 6.5 after Mellin separation. The proof of Proposition 5.2 writes $k=u_0sv^2$ and sums Lemma 6.6 over $v$.
• Section 7, Proof of the transfer proposition (pp. 29--35). Lemma 7.1 (first Poisson summation) reduces $\mathcal A(W)$ to the nonnegative form $\mathcal Q_{\xi_1}(U)$ (7.2) of Möbius sums; Lemma 7.2 (second Poisson identity, (7.6)) returns the Möbius coefficients to cubic Gauss-sum coefficients with a new column scale and exclusion; Lemma 7.3 bounds $\mathcal Q_{\xi_1}(U)\ll D^{\varepsilon_0}\Sigma(1+S_m)$ by regrouping the preimages, dyadic partition, Lemma 4.4 and Lemma B.2; the two lemmas give Proposition 5.4.
• Appendix A, Arithmetic identities and theta calculations (pp. 35--44). Proof of Lemma 4.1: Jacobi sums at a prime, $J(\chi_p^2,\chi_p^2)=-p$, the normalized quadratic Gauss sum (A.1) evaluated by Poisson summation with a Gaussian and its four square classes modulo $4\mathcal O$, cubic reciprocity. Proof of Proposition 6.2 and Lemmas 6.3, 6.4: finite Fourier expansion of the twists, reduced denominators and the choice of cusp, Kubota's automorphy law $\theta(g_1w)=\kappa(g_1)\theta(w)$ with $\kappa(g_1)=(c_1/a_1)_3$, the Mellin transform of the horizontal derivative, the functional equation (A.18), Phragmén--Lindelöf and contour shifts giving $V_*^\sharp$.
• Appendix B, Separating variables in smooth weights (pp. 44--46). Lemma B.1: Mellin separation of a smooth compactly supported kernel with coefficient bounds by finitely many derivatives; Lemma B.2: a mean-square bound for a common test function implies one for row-dependent kernels at the cost of a $C^{2m+4}$ norm; Lemma B.3: weighted Cauchy--Schwarz for recombining.
• References (pp. 46--49). Five listed entries (Gao--Zhao 2023, Lu, Zaman and Zhao 2026, Page 1935, Siegel 1935, Tatuzawa 1951) are not cited anywhere in the TeX text.

External inputs the proofs rest on, at statement level: Patterson's coefficient formula and cusp tables (Theorem 8.1, Table II and Table III of Patterson 1977) in the normalization of Dunn and Radziwiłł (their Section 5, Appendix A and Propositions 5.1--5.2, and (1.5) for the supplementary law for $\lambda$); Kubota's automorphy law; Goldmakher and Louvel's quadratic large sieve over number fields (Theorem 1.1 and Corollary 1.2); Landau's prime ideal theorem; Hecke's meromorphic continuation; Dirichlet's $L(1,\chi_{-3})>0$; and from Iwaniec--Kowalski the lattice Poisson formula (Theorem 4.5), the primitive Gauss-sum identity (3.12), the Gauss--Jacobi relation (3.18), Stirling's formula and Phragmén--Lindelöf (Theorem 5.53). The text flags nothing as numerical, computer-assisted or conditional; Corollary 1.2 claims an absolute effective constant. The only components the manuscript itself leaves unproved are Corollary 1.2 and the consequences paragraph of Section 1, each referred to the literature.

Bears on

The manuscript names no Erdős problem. Every row below is a proposed relation, the claim behind it is unverified here, and each page's status rests on its own acceptance evidence, not on this card.

• Problem 770: proposed input to question 3. The partial result the page records proves $h(n)=P(n)$ for $P(n)>n^\epsilon$ only for $\epsilon>1/2$; its criterion $C(p)>n$ needs $p>\sqrt n$. Section 1 of the manuscript states, with citations to Rodosskii and Montgomery--Vaughan's Theorem 13.12 and without proof, that Theorem 1.1 gives a bound of the form $(\log p)^{C}$ for the least quadratic nonresidue modulo an odd prime $p$; the manuscript states only the quadratic case. Any argument below $\epsilon=1/2$ would be a different argument, not made in the manuscript and not checked here. The claim is unverified; the page's status rests on acceptance evidence.
• Problem 985: proposed input only. The manuscript mentions neither primitive roots nor this question; its uniform zero-free half-plane for all Dirichlet $L$-functions (Theorem 1.1) and its uniform progression count (Corollary 1.2) are the kind of hypothesis under which a prime primitive root below $p$ has been approached, and the reading that they replace GRH in such an argument is this corpus's reading, not the paper's. Unverified; the page's status rests on acceptance evidence.
• Problem 969: proposed input only. A fixed zero-free half-plane for $\zeta(s)$ gives a power-saving bound for the Möbius sum (the manuscript's Section 3 proves the ideal-sum form (3.3) with exponent $11/12$), and an error term for the squarefree count below the $x^{1/2}$ barrier would follow through the series $\zeta(s)/\zeta(2s)$; the manuscript mentions neither the count of squarefree rational integers nor this question, and the deduction was not checked here. Unverified; the page's status rests on acceptance evidence.
• Problem 769: proposed input to an unaccepted proof claim. The page records a claimed upper bound for odd $n$ with the exponent $1/(4\sqrt e)$, which is the Burgess exponent for the least quadratic nonresidue, and $C(\log n)^2$ under GRH; that the manuscript's stated but unproved polylogarithmic nonresidue consequence could sharpen the unconditional exponent is an inference from the matching exponent alone, since the claim's write-up was not read here and the manuscript's consequence concerns prime moduli. The manuscript names neither the problem nor cube decompositions. Unverified; the page's status rests on acceptance evidence.
• Problem 1204: removal of a conditional obstruction. Granville's card, which bears on the page, records constructions that assume infinitely many Siegel zeros and would make $A(k)\sim k\log k$ fail; the manuscript claims (abstract and Section 1.1) that Theorem 1.1 rules out Landau--Siegel zeros, which would leave those constructions with a false hypothesis. It proves nothing about $A(k)$ or $B(k)$ and changes no status. Unverified; the page's status rests on acceptance evidence.
• Problem 855: the same conditional obstruction, through the same card; the manuscript says nothing about $\pi(x+y)\le\pi(x)+\pi(y)$ or primes in short intervals (Corollary 1.2 has error $x^{11/12}\log x$, far above any short-interval scale). Unverified; the page's status rests on acceptance evidence.
• Partial collective-coprimality threshold bound: the card's result stops at $\epsilon>1/2$ (its criterion $C(p)>n$ needs $p>\sqrt n$); the manuscript's stated (cited, unproved in the text) polylogarithmic bound for the least quadratic nonresidue would be an input to a different argument below $\epsilon=1/2$, not made in the manuscript and not checked here. Unverified.
• Sieving intervals and Siegel zeros: the card's Corollary 3, Proposition 2 and the conditional consequence for $A(k)$ assume infinitely many Siegel zeros; the manuscript claims to exclude Landau--Siegel zeros, so if its claim stands those results have a false hypothesis and no unconditional content. Unverified.
• Estimates for representation numbers of quadratic forms: the card's Theorem 5 has a sharper form "when there are no Siegel zeros"; the manuscript's claim would make that form unconditional. The problem that card bears on is already disproved, so no status is affected. Unverified.
• The maximal order of the shifted-prime divisor function: the card's unconditional good-moduli page imports an exceptional-zero statement (at most one primitive character of conductor below $V$ with a zero in $\operatorname{Re}s>1-1/W$, $|\operatorname{Im}s|\le V$) and Montgomery's zero-density count of the characters with a zero in that region, where $W=((2/5)\log x)^{3/4}$; once $W>12$ the region lies inside $\operatorname{Re}s>11/12$, so if Theorem 1.1 holds both the exceptional-conductor deletion and the zero-density count become vacuous for large $x$. The numerical constant $\theta=0.4736$ and the card's GRH-conditional branch are unchanged, the latter not reached by a half-plane at $11/12$. Unverified.