Improved zero-free half-planes for the quasi-Riemann hypothesis

Baiying Liu

REVIEW

Improved zero-free half-planes for the quasi-Riemann hypothesis

Baiying Liu

Number Theory · Submitted 8 Oct 2026

Pith's verdict: A careful parameter re-optimization that shaves 7/8 by 4e-5, with exact algebraic control and honest limitations—but every deep input still rides on an unpublished preprint.

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arXiv ID2610.12234 (version 2)

Pith IDDWA3NFIL, Pith's permanent record of this paper and its review

CitationBaiying Liu. "Improved zero-free half-planes for the quasi-Riemann hypothesis." arXiv:2610.12234 (2026). https://pith.science/paper/DWA3NFIL

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Editor's note

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Read it if you care about the OpenAI quasi-RH preprint. The paper does something genuinely finicky: it varies the two length parameters in the 7/8 argument and shows the boundary can be moved by a controlled, exact amount. The new result is not just the number 0.874957; it is the proof that the move is governed by an algebraic identity whose discriminant produces sqrt(921). Lemma 24.1 is the heart and the algebra checks out. Proposition 25.1 gives a clean weighted-cancellation lower bound under three explicit affine constraints, and Lemma 19.1's Bernstein-basis verification is a legitimate exact computation, with a claimed checker and Lean files named. The parameters are located by balancing equations, not fit to the target, and the paper says so itself. It also states its limitations straight: the improvement is small, balancing alone is insufficient, optimality is an expectation, and varying kappa could give another 4.5e-8. The internal computations look right. I spot-checked the parameter values, the derivative bound 45/32, the margin -19/320000, the weights in Prop 25.1, and the 14-row Bernstein partition. The endpoint inequality is proved over the full rectangle by an exact identity, so this is not a curve-fit. The soft spot is the import chain, and it is load-bearing. The unconditional main theorem inherits every deep analytic input from an unpublished preprint [1] with no URL or venue: beta* <= 7/8, the reflected mean square, the unweighted fourth moment at fixed kappa=3/4, and the joint local error. Section 18 asserts the moment hypotheses hold, and Lemma 22.1 re-derives the reflected estimates including m+l<1 cases, but none of these inputs can be checked independently. If one fails at the new parameters, the exact endpoint inequality proves nothing. The paper acknowledges the dependency—honest, but it does not make the theorem verifiable. The Lean/checker claim is similarly hard to use: file paths and namespaces are given, but no repository URL or commit hash. There is also a minor rendering inconsistency in b_new in the abstract, disambiguated by the stated orientation value. Who is this for: anyone working on the quasi-RH line, and anyone who wants a model of exact parameter optimization over an existing estimate system. The added value over [1] is small, but the new inequalities are exactly checkable and that is real. Recommendation: send it to a serious referee, but only after [1] is released or published and the Lean repository is pinned. The referee should verify the import chain at the new parameters, not just Lemma 24.1.

Referee report

Significance

Conditional on the imported estimates of [1] remaining valid at the shifted parameters, this is an honest, incremental advance: it shows that the parameters of [1] are not extremal, without fitting parameters to the target. The algebraic bound is located by the balancing system R* = 2/3, E = 0, ∂_δ E = 0 at the critical class, with √921 entering through the discriminant of 288δ² − 588δ + 185; Lemma 24.1 then proves the endpoint inequality on the full rectangle with equality only there. Proposition 25.1 is a genuinely parameter-free lower bound: positive weights cancel M and ℓ, so under the three affine constraints every admissible boundary satisfies B ≥ B_new. The arithmetic content is machine-checked: the Bernstein minima of Lemma 19.1, the discriminant identity (24.4), and the rational sign brackets are verified by the exact-arithmetic checker in [15], and the paper reports Lean 4 formalizations with transitive axioms only propext, Classical.choice and Quot.sound. System-level optimality is explicitly an expectation, and Section 15's statement that Parts II–III 'use the proved intermediate results of [1]' flags the conditional character of the theorems.

Major objections (2)

[§15–§18, Props. 18.1–18.2, Lemmas 9.3/16.1/22.1; ref. [1]] The two main theorems (Theorems 15.1 and 21.1) are stated unconditionally, but every deep analytic input is imported from the unpublished preprint [1], and the manuscript asserts rather than verifies that the imported hypotheses hold at the shifted parameters. Proposition 18.1 fixes κ = 3/4 in the unweighted fourth-moment estimate (Lemma 9.3 = [1, Lemma 18.1]); with z_p > 0 and κ < 1 that lemma requires the premise β* ≤ (1+κ)/2, which is supplied only by the imported parent bound β* ≤ 7/8 (Theorem 13.1, from [1]). Section 18 states 'All moments below have exactly the common-character, coefficient and smooth-weight hypotheses of Section 9', and Section 10 states 'The required moment hypotheses therefore hold', without checking, for example, the length condition n_1+n_2+6κz_p ≤ M_Θ and the mesh restriction of Lemma 9.3 at ℓ_new = (33+8√921)/1653 > 1/6 and b_new = −4/29 + 230√921/26709; Lemmas 16.1 and 22.1 similarly take the reflected mean square [1, Lemma 14.3] as given in ranges where m+l−1 has either sign, and Proposition 18.2 invokes 'the unchanged joint-error and moment hypotheses of Part I' (i.e., [1, Proposition 16.1]). Because [1] is a preprint with no public URL and none of these estimates is re-derived here, the margin 7/8 − B_new = 4.293 × 10⁻⁵ is not protected: a hypothesis violation or an error of size greater than this margin in any imported exponent at the new parameters would reverse the sign of the central inequality E_B(h) ≤ 0. The revision should either verify each imported lemma's displayed hypotheses at (b_new, ℓ_new, κ = 3/4) explicitly, or restate the main theorems as conditional on a precise, closed list of imported statements that this paper does not itself prove.
[References [15]; abstract] The abstract states 'This result has been formalized by Lean ([15])', and reference [15] reports public Lean conclusions beta_star_le, hecke_nonzero, dirichlet_nonzero and zeta_nonzero with transitive axioms only propext, Classical.choice and Quot.sound. This claim is in tension with the verification gap described above: either the formalization re-derives the imported estimates of [1] inside Lean, which would close the gap and should be stated precisely; or the formalized statements are conditional on hypotheses supplied as parameters rather than axioms, in which case their content is weaker than the abstract suggests; or only the algebraic lemmas (Lemma 19.1, Lemma 24.1, Proposition 25.1) are formalized, in which case the formalization does not certify the analytic chain. Reference [15] lists file paths and namespaces but not the exact statement of each formalized theorem nor the status of [1]'s results within the formalization. The revision should add a precise scope note that maps each formalized declaration to the corresponding displayed theorem and states which analytic inputs are assumed, proved, or supplied as parameters inside the Lean development.

Minor objections (6)

[References [1]] Reference [1] gives no URL or repository identifier ('Preprint, 30 September version. Public manuscript.'); since the entire analytic content of the paper is imported from this source, a stable identifier and a versioned link are required for the claims to be checkable.
[Title page / affiliation] The affiliation line contains a typo: 'West Laf ayette' should read 'West Lafayette'.
[References [15]] The sentence 'The repository for this paper and its accompanying Lean files is GitHub repository [15]' is redundant, and the entry for [15] should include the repository URL and the formalization version stamp in addition to the Lean version.
[Section 2 (symbol table)] The symbol table in Section 2 omits the row-norm exponent d, the subset length d_I, and the critical zero-class parameter δ_c, all of which appear later; the remark in Section 16 that 'd_I is unrelated to the norm exponent d' is helpful, but these symbols belong in the table.
[Lemma 19.1 (covering table)] The covering rectangles in the Bernstein-basis table of Lemma 19.1 overlap; since the coverage claim is machine-verified, a sentence indicating that the overlaps are deliberate and that the listed rectangles together cover [0,5/6] × [0,1/2] would make a hand-check easier.
[Sections 3 and 12] The comparisons 'The two applications in Part I use σ0 C(s) C(σ0) 11/12 s−2/3 1/4 7/8 s−11/16 3/16' (Section 3) and the parameter table in Section 12 are set as inline text; typesetting them as formal tables would improve readability.

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The paper

The paper claims

Every finite-order Hecke $L$-function over $\mathbb{Q}(\sqrt{-3})$, and every Dirichlet $L$-function including $\zeta$, is zero-free beyond real part $0.874957$, a slight sharpening of $7/8$ by balancing two length parameters in the…

The reading

The quasi-Riemann hypothesis asks for one fixed line $\mathrm{Re}\, s = B$ with $B < 1$ that clears all nontrivial zeros of the Riemann zeta function, uniformly across a family of $L$-functions. This paper claims that the uniform zero-free half-plane for finite-order Hecke $L$-functions over the Eisenstein field $\mathbb{Q}(\sqrt{-3})$ — and, by transfer, for every Dirichlet $L$-function including $\zeta$ — can be pushed from the established $7/8$ to $B_{\mathrm{new}} = (1507 - 2\sqrt{921})/1653 \approx 0.874957$. The mechanism is to treat two length parameters of the underlying cubic-$\theta$ averaging argument, the summation-exponent difference $b$ and the total auxiliary prime length $\ell$, as free variables and balance them at a critical zero class; a rational variant with $b$ fixed gives $B_r = 34999/40000$. A closing algebraic statement (Proposition 25.1) shows that under three explicit affine constraints of the same estimate system no choice of lengths reaches below $B_{\mathrm{new}}$. Both new bounds are reported to carry a machine-checked formalization.

What carries the argument

The load-bearing object is the modified cubic-$\theta$ average $J_\eta(Z)$: the reflection principle bounds it directly with exponent $l_x/2 + b/12$, while Poisson summation isolates the principal Mellin integral with exponent $C_b(s) = s - (4 + b)/6$, and matching the two exponents at the boundary $B(\ell) = 11/12 - \ell/4$ determines the zero-free line. Two length scales enter, the element-scale exponents $l_x, l_y$ with difference $b$, and the total logarithmic length $\ell$ of selected auxiliary primes; the competition between lengthening $\ell$ and controlling nonprincipal terms is encoded in the relative exponent $E(b, \ell; \delta, x)$, with $\delta$ the zero-class parameter and $x$ the normalized prime-sum amplitude, at the fixed moment value $\kappa = 3/4$ that the parent bound $\beta^* \le 7/8$ makes legal. The rational part verifies $E \le -19/320000$ by an exact calculation in the Bernstein basis polynomials (nonnegative partition-of-unity polynomials) on a rectangle partition. The algebraic part pivots on the character-count exponent $R^*(\delta, 1/2) = (185 - 288\delta + 96\delta^2)/(185 - 138\delta)$: the class $R^* = 2/3$ forces $\delta_c$, then $\ell_{\mathrm{new}}$ and $b_{\mathrm{new}}$, and the endpoint inequality is settled by the exact identity $4A(y)Q = (2A(y)\delta + D(y))^2 + y(d_1 + d_2 y + d_3 y^2 + d_4 y^3) \ge 0$, with equality only at $(\delta_c, 1/2)$. The optimality statement Proposition 25.1 combines the three affine constraints $B \ge L_0$, $B \ge L_K$, $B \ge H_c$ with positive weights that sum to one and cancel the free lengths, leaving $B \ge B_{\mathrm{new}}$ for every admissible choice.

Extended reading notes

Core claim

The central claim is Theorem 21.1: the family supremum $\beta^*$ over real parts of nontrivial zeros of all primitive finite-order Hecke $L$-functions over $F = \mathbb{Q}(\sqrt{-3})$ satisfies $\beta^* \le B_{\mathrm{new}}$, and consequently every finite-order Hecke $L$-function over $F$, every Dirichlet $L$-function, and $\zeta$ itself is zero-free on $\mathrm{Re}\, s > B_{\mathrm{new}}$, with principal poles at $s = 1$ allowed. Numerically $7/8 - B_{\mathrm{new}} = 4.293 \times 10^{-5}$, but the improvement is structural rather than numerical: the parent $7/8$ argument fixed the parameters at $b = 1/8$, $\ell = 1/6$, while the balanced comparison boundary $B(\ell) = 11/12 - \ell/4$ shows that increasing $\ell$ lowers the boundary, subject to the nonprincipal error exponent $E(b, \ell; \delta, x)$. The new parameters are forced by the critical zero class where the character-count exponent $R^* = 2/3$ cannot be improved by changing $b$: $\delta_c = (49 - \sqrt{921})/48$ is the root of $288\delta^2 - 588\delta + 185 = 0$, $\ell_{\mathrm{new}} = (33 + 8\sqrt{921})/1653$ solves $E = 0$, and $b_{\mathrm{new}} = -4/29 + 230\sqrt{921}/26709$ solves stationarity in $\delta$; the $\sqrt{921}$ is the discriminant of that quadratic, not a free choice. At those parameters the endpoint inequality is an exact quadratic identity with equality only at the critical pair, and Proposition 25.1 proves that under the three explicit affine constraints of the same estimate system every admissible boundary must satisfy $B \ge B_{\mathrm{new}}$. The intermediate rational bound $B_r = 34999/40000$, obtained by moving $\ell$ to $1/6 + 1/10000$ with $b$ fixed, is verified by an exact sign calculation in the Bernstein basis, and the paper reports machine-checked formalizations of both bounds.

Load-bearing premise

The argument leans on the unpublished preprint [1] for every imported moment and error estimate, and those estimates — the unweighted fourth moment at $\kappa = 3/4$, the reflected $\theta$ mean-square, the joint local error, and the parent bound $\beta^* \le 7/8$ — are asserted, not re-derived, at the shifted parameters $b_{\mathrm{new}}, \ell_{\mathrm{new}}$; if any one of them fails there, the endpoint inequality $E \le 0$ does not yield the theorem.

Editorial extensions

If this is right

Every finite-order Hecke L-function over Q(√−3) and every Dirichlet L-function, including ζ, is zero-free for Re s > B_new ≈ 0.874957, with principal poles at s = 1 allowed; the gap does not narrow with conductor or height.
Under the three explicit affine constraints of the estimate system, Proposition 25.1 makes B_new the exact endpoint: every admissible choice of free lengths gives B ≥ B_new, with equality at the new lengths; the paper labels broader optimality across the full system an expectation rather than a proved statement.
Fixing the moment parameter at κ = 3/4, which the parent bound β* ≤ 7/8 makes legal, removes the loss carried by the original varying-moment argument, and the strict comparison margins close in a Mellin-continuation contradiction on the new line.
The paper notes that letting κ range outside [3/4, 1] could shave about 4.48 × 10⁻⁸ more off B_new, an option left to future work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The balancing recipe is general: applied to the analogue of the critical class R* = 2/3 in another quadratic base field, or to a different metaplectic theta ladder, the same two-parameter optimization should produce a distinct algebraic endpoint of the same type.
The single most exposed step is the assertion that the moment and error estimates imported from [1] hold at the shifted parameters; a direct numerical check of the unweighted fourth moment at κ = 3/4 at the new scales would test exactly that.
The tiny size of the gain suggests the practical ceiling of the method lies in the moment and reflected mean-square inputs rather than in the exponent balancing; improving those inputs uniformly would convert immediately into a new boundary via the same balance equations.

Original abstract

Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [Ope26a]. In its proof, there are several parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$, and $\kappa$. In this paper, via varying $b$, $\ell$, and $\kappa$, following the same strategy, we slightly improve the bound $\frac{7}{8}=0.875$ to \[ B_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+7}{18\xi_{\mathrm{new}}+9} =0.874957019420098946\ldots, \] where $\xi_{\mathrm{new}}$ is the unique root in $(0.38858,0.38859)$ of $1188\xi^3-2160\xi^2+171\xi+190=0$. Explicitly, with $\arccos\in[0,\pi]$, \[ \xi_{\mathrm{new}}=\frac{20}{33}+\frac{\sqrt{1391}}{33} \cos\!\left(\frac13\arccos\frac{28470}{1391^{3/2}} -\frac{2\pi}{3}\right). \] The limiting parameters are \[ \begin{gathered} b_{\mathrm{new}}=\frac{8(3\xi_{\mathrm{new}}-1)(12\xi_{\mathrm{new}}^2-40\xi_{\mathrm{new}}+15)} {9(2\xi_{\mathrm{new}}+1)(24\xi_{\mathrm{new}}^2-20\xi_{\mathrm{new}}+5)}, \ell_{\mathrm{new}}=\frac{5-6\xi_{\mathrm{new}}}{9+18\xi_{\mathrm{new}}},\qquad \kappa_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+5}{18\xi_{\mathrm{new}}+9} =2B_{\mathrm{new}}-1. \end{gathered} \] We also prove its optimality within the explicitly specified exponent comparison system. This result has been formalized by Lean [Liu26].

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-09: 1 request, 4,047 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.13 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.95 · mentions code or data availability 0.28.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [1]. In its proof, there are two chosen parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$. In this note, via varying $b$ and $\ell$, following the same strategy, we slightly improve the bound $\frac{7}{8}=0.875$ to be $$B_{\mathrm r}=\frac{34999}{40000}=0.874975, (\text{when }b_{\mathrm r}=\frac{1}{8}, \ell_{\mathrm r}=\frac{1}{6} + \frac{1}{10000});$$ and \[ \begin{gathered} B_{\mathrm{new}}=\frac{1507-2\sqrt{921}}{1653} =0.874957069799\ldots, (\text{when }b_{\mathrm{new}}=-\frac{4}{29}+\frac{230\sqrt{921}}{26709}, \ell_{\mathrm{new}}=\frac{33+8\sqrt{921}}{1653}). \end{gathered} \] This result has been formalized by Lean ([15]).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [1].	background 1.00	0.03	0.06	asserted 1.00
s1	In its proof, there are two chosen parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$.	background 0.88	0.09	0.02	asserted 1.00
s2	In this note, via varying $b$ and $\ell$, following the same strategy, we slightly improve the bound $\frac{7}{8}=0.875$ to be $$B_{\mathrm r}=\frac{34999}{40000}=0.874975, (\text{when }b_{\mathrm r}=\frac{1}{8}, \ell_{\mathrm r}=\frac{1}{6} + \frac{1}{10000});$$ and \[ \begin{gathered} B_{\mathrm{new}}=\frac{1507-2\sqrt{921}}{1653} =0.874957069799\ldots, (\text{when }b_{\mathrm{new}}=-\frac{4}{29}+\frac{230\sqrt{921}}{26709}, \ell_{\mathrm{new}}=\frac{33+8\sqrt{921}}{1653}).	result 0.82	0.94	0.93	asserted 1.00
s3	\end{gathered} \] This result has been formalized by Lean ([15]).	result 0.36	0.32	0.06	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes
math.NT 2026-10 conditional novelty 7.0 of 10

Under a quasi-Riemann hypothesis with zero real parts at most Theta < 11/12, the exceptional sets in the binary Goldbach and prime-difference problems are O(N^{1/2} (log N)^{100}).
Formal Proof of an Anisotropic Calder\'on Uniqueness Theorem
math.AP 2026-11 accept novelty 5.0 of 10 full

A Lean 4 formal proof of a restricted anisotropic Calderón uniqueness theorem, checked by a comparator, is announced.
Zero-free half-planes for Hecke $L$-functions over number fields and Quasi-Riemann Hypothesis
math.NT 2026-10 conditional novelty 5.0 of 10

Every finite-order Hecke L-function over every number field is claimed zero-free for Re(s) > 0.874886647..., improving the previous uniform boundary 31/32.
Amplified moments and a refinement of the quasi-Riemann hypothesis
math.NT 2026-10 unverdicted novelty 5.0 of 10

Assuming the analytic inputs of [OAI26], the zero-free half-plane for finite-order Hecke L-functions over Q(sqrt(-3)) and for Dirichlet L-functions is refined from 7/8 to B1 = 8748867/10000000.
Primes between 8th powers?
math.NT 2026-10 conditional novelty 4.0 of 10

Assuming a claimed zero-free half-plane Re(s)>7/8 for the Riemann zeta function, the paper derives explicit prime-containing intervals around eighth powers and a Hoheisel-style interval, all conditional on that hypothesis.

Discussion

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