Amplified moments and a refinement of the quasi-Riemann hypothesis

Hejian Sang; Yuanda Xu

REVIEW

Amplified moments and a refinement of the quasi-Riemann hypothesis

Yuanda Xu, Hejian Sang

Number Theory · Submitted 11 Oct 2026

Pith's verdict: Careful, transparent conditional refinement: the new averaged-numerator machinery looks sound, but the theorem carries all of [OAI26]'s risk.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 14, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2610.15291 (version 2)

Pith IDUT4IAIBJ, Pith's permanent record of this paper and its review

CitationHejian Sang, Yuanda Xu. "Amplified moments and a refinement of the quasi-Riemann hypothesis." arXiv:2610.15291 (2026). https://pith.science/paper/UT4IAIBJ

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Short version: this is a careful, transparently conditional paper, and UNVERDICTED is the right call. It takes the OpenAI 7/8 preprint [OAI26] as a black box — explicitly, in Section 2 and in the AI-use disclosure — and refines the boundary to 8748867/10000000 by averaging the reflected numerator over the detected row class instead of bounding it pointwise. The theorem is only as secure as [OAI26], and the paper says so in so many words.

What is actually new is worth naming. Proposition 5.1 (the amplified first moment of the reflected numerator, using two copies of a plain block plus selected primes in the fourth moment), Proposition 3.1 (retaining the external decay in the low estimate, which lets the total prime length exceed 1/6), and Lemma 2.1 (extending the plain-moment range to κ ≥ 13/18) are real additions, not relabelings. The two-stage licensing chain 7/8 ⇒ B0 ⇒ B1 is not circular: the first stage proves β* ≤ B0, and only then is the smaller κ used. Lemma 8.1, which turns a licensed admissible choice into a zero-free boundary, is clean and short.

The verification material is genuinely checkable. Appendix D does not sample the parameter rectangle; it clears denominators and reduces the endpoint bound to positivity of polynomial coefficients, which holds on the whole rectangle. That is the right way to certify this kind of numerical claim, and it can be verified in an afternoon. Appendix G lists the imported inputs one by one, and the disclosure states flatly that the [OAI26] proofs were not independently verified here.

Soft spots, in proportion. The controlling risk is external: if any imported moment or the 7/8 half-plane fails, Theorem 1.1 falls with it. That is acknowledged, not concealed, so the honest question for a referee is whether the new parts are right under the stated hypotheses. The place I would look hardest is Lemma 2.1: it claims the κ-extension only modifies numerical bounds in the source finite induction, and since that induction is unverified, the claim cannot be fully checked without reading [OAI26] itself. The gain is also numerically small — about 7e-5 over Liu26's boundary, and the second stage adds only 8e-9 over B0 — and the parameters are tuned to make the boundaries close. That is normal in this kind of optimization, but it means there is no slack in the argument.

Recommendation: send it to a serious referee. The internal parts look coherent, the certificates are concrete, and the conditionality is stated plainly enough that a referee knows exactly what to check. I would not cite the boundary as established until [OAI26] clears review, but I would read the new moment argument if I worked in this area.

Referee report

Significance

If the analytic inputs of [OAI26] are correct, the paper is a substantial and carefully executed refinement: it gives a concrete numerical improvement over both the source 7/8 boundary and the value BL of Liu, and it introduces several reusable ingredients, notably the low-side decay retention, the averaged numerator estimate in Proposition 5.1, and exact quadratic-endpoint certificates in Appendix D. The two-stage licensing argument is internally coherent and is not logically circular, as the authors are explicit that each bound is used only after it has been proved. The manuscript is unusually transparent about its dependence on [OAI26], including an explicit disclosure that the source proofs have not been independently verified. The main significance, however, is conditional: the paper establishes an implication from a large body of unverified inputs, not an unconditional zero-free half-plane.

Major objections (3)

[§1.2, §2, AI use disclosure] Theorem 1.1 is conditional on the analytic inputs of the unpublished preprint [OAI26], including the marked inverse moment (4), the sixth-power amplification (5), the plain fourth moment (6), the simultaneous witnesses (9), and the family-wide 7/8 zero-free half-plane. The paper acknowledges this explicitly, most clearly in the AI use disclosure: 'The resulting zero-free conclusions are conditional on the correctness of the analytic inputs from [OAI26], whose proofs have not been independently verified here.' This conditionality is load-bearing because the boundary B1 stands or falls with those inputs. I found no internal contradiction in the new part of the argument, but the central claim of the manuscript cannot be certified independently of the source preprint. The editor will need to decide whether theorems conditional on an unpublished arXiv preprint are acceptable for this journal; if so, the conditional status should be kept as prominent in the theorem statement and abstract as it is in the disclosure. The authors acknowledge this: “The resulting zero-free conclusions are conditional on the correctness of the analytic inputs from [OAI26], whose proofs have not been independently verified here. 20 A The extension of the plain-moment range This extension is needed only for the second stage.”
[Appendix A, Lemma 2.1] The second stage uses kappa1 = 29991/40000, which is smaller than the source's stated range kappa >= 3/4 for the plain fourth moment. Lemma 2.1 therefore supplies a necessary extension of the moment range, and its proof in Appendix A is an adaptation of the proof of [OAI26, Lemma 18.1]. The checks of the parameter-dependent steps are detailed, but the induction skeleton, its numerical reserves, and the 'same finite induction' invoked in the closing paragraph belong to the unpublished source. Because this extension is needed for Theorem 1.1 and cannot be checked from the information supplied in this manuscript, a referee cannot currently certify the second stage on the basis of the text alone.
[§3 and Appendix G] The proof of Proposition 3.1 depends on structural facts imported from [OAI26], in particular the two branches of [OAI26, Lemma 14.3] and the relation M' + ell' - 1 = -3d. The manuscript states these as the starting point of its low-side calculation rather than proving them. This is a legitimate use of hypotheses if [OAI26] is accepted as a source, but since that source is unpublished, the low-side decay retention that is one of the paper's two named improvements is not verified within the manuscript itself. I would not raise this as a technical error; it is a further consequence of the same external-dependence issue as the previous comment.

Minor objections (4)

[Throughout] There are numerous missing spaces in the typeset text, for example 'HeckeL-functions' in the abstract and inline fractions such as '874886710000000'; a pass with a proper LaTeX/amsthm source is needed.
[§1.2] The proof table lists B0 = 69991/80000 and B1 = 8748867/10000000, but the displayed values are easy to misread because of missing separators in the inline fractions; separate the numerator and denominator for clarity.
[§1.2 and Theorem 1.1] The abstract says 'we improve the boundary', which could be read as an unconditional statement. Since the main theorem is explicitly conditional on [OAI26], the abstract and opening sentence should say 'conditionally improve' or otherwise make the conditional status visible in the first sentence.
[Appendix G and references] The reference [OAI26] is an unpublished preprint with a dated version and a GitHub URL. For reproducibility, please also provide a stable identifier or a freeze of the exact version cited, and state clearly whether all imported statement numbers refer to that frozen version.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Assuming the predecessor work's moment inputs, this paper pushes the zero-free boundary for Hecke and Dirichlet L-functions from 0.875 to 0.8748867.

The reading

Conditionally on the analytic inputs of the predecessor preprint [OAI26] — above all its uniform $7/8$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$ and its three moment estimates — this paper proves that every such Hecke $L$-function, and by transfer every Dirichlet $L$-function including $\zeta(s)$, has no zeros in $\operatorname{Re}(s) > B_1 = 8748867/10000000 = 0.8748867$. Since a zero of $L(s,\eta)$ would be a pole of its reciprocal $1/L(s,\eta)$, the conclusion lets each reciprocal extend holomorphically through that half-plane, and a corollary bounds weighted smoothed Möbius sums by $X^{B_1+\varepsilon}$. The gain over the previous comparison $B_L = 0.87495707\ldots$ is only about $7\times 10^{-5}$, but it is achieved without strengthening the inverse-moment estimate: a relaxed low estimate retains a decay factor that earlier versions discarded, and an amplified first moment averages the reflected numerator over the detected row class instead of bounding it pointwise. The two stages are ordered so that each new moment parameter is used only after the boundary licensing it is proved: $7/8$ licenses $\kappa = 3/4$, giving the intermediate boundary $B_0 = 69991/80000$, which licenses $\kappa = 29991/40000$, giving $B_1$. The author states in the disclosure that the conclusions are conditional on inputs whose proofs were not independently verified there.

What carries the argument

Three mechanisms carry the argument. First, the low-side refinement (Proposition 3.1) keeps the decay of each rescaled term of the compensated probe $I_\eta(Z)$ — the finite alternating sum over prime windows and subsets of slots — until after the completed-row norm is bounded, so the total prime length $\ell$ may exceed $1/6$; inequality (17) is what absorbs the longer windows. Second, the principal new analytic step, Proposition 5.1, is an amplified first moment of the reflected numerator: instead of a pointwise bound, the numerator is averaged over a detected row class using the same physical prime polynomials that defined the class, with the plain fourth moment applied through an approximate functional equation; the balance condition $-\vartheta m - qv(1-2m)/2 = -\vartheta/2$ makes the saving independent of block length. Third, the endpoint certificate (Proposition 7.1) reduces the high-side comparison to checking that a numerical function $E_{b,\ell,\kappa}(\delta,x)$ is negative on a rectangle, which is done by completing the square of an exact quadratic with positive coefficients; Lemma 8.1 then converts any licensed admissible parameter triple into a zero-free boundary through the reciprocal-pole contradiction.

Extended reading notes

Core claim

The central claim, Theorem 1.1, is that if the analytic inputs of [OAI26] hold — the family-wide $7/8$ zero-free half-plane, the marked inverse moment, the sixth-power amplification, the plain fourth moment, and the simultaneous-witness lemma — then every finite-order Hecke $L$-function over $F = \mathbb{Q}(\sqrt{-3})$, and by norm-pullback every Dirichlet $L$-function including $\zeta(s)$, has no zeros in $\operatorname{Re}(s) > B_1 = 8748867/10000000$. The proof runs the licensing chain $\beta^* \le 7/8 \Rightarrow \kappa = 3/4 \Rightarrow \beta^* \le B_0 = 69991/80000 \Rightarrow \kappa = 29991/40000 \Rightarrow \beta^* \le B_1$, where $\beta^*$ is the supremum of the real parts of the family's zeros and $\kappa$ is the plain-moment parameter; Section 2 is explicit that imported results are hypotheses of the theorem, and the ordering ensures the second $\kappa$ is used only after $B_0$ has been proved. Each stage is closed by Lemma 8.1: if a zero lay to the right of the claimed boundary, the reciprocal $1/L(s,\eta)$ would both extend holomorphically past that line, by the low and high estimates of the same compensated probe, and have a pole there, a contradiction. The boundary improves the previous value $B_L = 0.87495707\ldots$ by about $7\times 10^{-5}$, and Corollary 9.1 converts the widened half-plane into the bound $O(X^{B_1+\varepsilon})$ for smoothed Möbius sums with Hecke character weights.

Load-bearing premise

The entire conclusion rests on the correctness of the analytic inputs of [OAI26] — above all the uniform $7/8$ zero-free half-plane and the three moment estimates — which this paper assumes rather than proves, a point the author states openly in the disclosure, where the zero-free conclusions are described as conditional on inputs whose proofs were not independently verified here.

Editorial extensions

If this is right

The reciprocal $1/L(s,\eta)$ extends holomorphically to $\operatorname{Re}(s) > B_1$ for every fixed target in the family, with a saving that is common to all targets even though constants may depend on the character.
Weighted smoothed Möbius sums $\sum_a \mu(a)\eta(a)W(\operatorname{N}a/X)$ over $\mathbb{Q}(\sqrt{-3})$ are $O(X^{B_1+\varepsilon})$ (Corollary 9.1), improving on what the $7/8$ half-plane would give.
The zero-free half-plane transfers to all Dirichlet $L$-functions, including $\zeta(s)$, with the pole at $s=1$ for principal characters allowed.
The two-stage chain exhibits the boundary as its own license: each improved half-plane permits a smaller moment parameter $\kappa$, and the second stage's final move is worth $B_0 - B_1 = 8/10^7$.
The pole-vs-holomorphy clash that closes each stage is uniform: it is the contradiction that lets one half-plane improvement license the next without redoing the probe machinery.

Reading between the lines

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

Further iteration appears possible in principle: Lemma 2.1 keeps the plain moment valid down to $\kappa = 13/18$, while the two stages used $\kappa = 3/4$ and $\kappa = 29991/40000$, so the licensing chain has parameter room left before the moment range binds; the paper itself makes no claim about how far this can go and explicitly disclaims optimality among variants of the method.
The averaged-numerator move is not tied to the specifics of $\mathbb{Q}(\sqrt{-3})$: any family where a fourth moment of plain polynomials is known but the inverse moment is the bottleneck should admit the same replacement of a pointwise numerator bound by a moment average, though the paper only works out the cubic-field case.
If the inputs of [OAI26] are ever verified independently, Theorem 1.1 and Corollary 9.1 become unconditional without further work in this paper, making $\sigma \approx 0.8748867$ a concrete benchmark for the quasi-Riemann hypothesis over these families.

Original abstract

Recently, OpenAI announced a uniform $7/8$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$, together with its transfer to Dirichlet $L$-functions. Assuming the analytic inputs of that work, we improve the boundary from $7/8=0.875$ to $8748867/10000000=0.8748867$ for the same families. The improvement combines a relaxed direct estimate with an amplified first moment of the reflected numerator, obtained from the existing fourth-moment estimate. A two-stage parameter argument yields the stated boundary without strengthening the inverse-moment estimate.

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, 2026-10-14: 1 request, 3,704 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.53 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Recently, OpenAI announced a uniform $7/8$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$, together with its transfer to Dirichlet $L$-functions. Assuming the analytic inputs of that work, we improve the boundary from $7/8=0.875$ to $8748867/10000000=0.8748867$ for the same families. The improvement combines a relaxed direct estimate with an amplified first moment of the reflected numerator, obtained from the existing fourth-moment estimate. A two-stage parameter argument yields the stated boundary without strengthening the inverse-moment estimate.

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, OpenAI announced a uniform $7/8$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$, together with its transfer to Dirichlet $L$-functions.	background 1.00	0.03	0.05	asserted 0.99
s1	Assuming the analytic inputs of that work, we improve the boundary from $7/8=0.875$ to $8748867/10000000=0.8748867$ for the same families.	result 0.90	0.93	0.90	asserted 0.97
s2	The improvement combines a relaxed direct estimate with an amplified first moment of the reflected numerator, obtained from the existing fourth-moment estimate.	method 0.99	0.84	0.07	asserted 1.00
s3	A two-stage parameter argument yields the stated boundary without strengthening the inverse-moment estimate.	method 0.93	0.87	0.12	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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 2 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}).
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.

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.