Zero-free half-planes for Hecke $L$-functions over number fields and Quasi-Riemann Hypothesis

Baiying Liu

REVIEW

Zero-free half-planes for Hecke $L$-functions over number fields and Quasi-Riemann Hypothesis

Baiying Liu

Number Theory · Submitted 19 Oct 2026

Pith's verdict: A serious conditional result: clean scalar mechanics and a real new boundary, but the headline theorem currently leans on an unpublished preprint and on sections I could not see.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 20, 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 →

· 2 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 ID2610.26835 (version 2)

Pith IDSXEO7LTN, Pith's permanent record of this paper and its review

CitationBaiying Liu. "Zero-free half-planes for Hecke $L$-functions over number fields and Quasi-Riemann Hypothesis." arXiv:2610.26835 (2026). https://pith.science/paper/SXEO7LTN

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

Here is the short version. The paper has a genuine new boundary, Bflat = 0.874886647..., claimed uniformly over every number field, which improves the 31/32 of Cureton-Thorner and the 7/8 of Ope26a. It also has a clean, checkable scalar core. But the version I saw does not by itself support the headline theorem: the analytic engine comes from an unpublished preprint, and the sections that actually prove the refined boundary were not available to me. I would not desk-reject it, and I would not accept the main theorem on this version either.

What is genuinely good: the quartic root evaluates correctly, Appendix B gives a genuine exact-arithmetic rational chain, and the Bernstein-basis inequalities in Section 2 are explicit and checkable. I found no internal error in the visible Sections 1 through 12. The paper is also honest about its own status: the abstract and Remark 3.1 name [Ope26b] as the source of the exceptional theta datum, local Whittaker table, and radial functional equation, and the acknowledgments say the Lean formalization is incomplete. That disclosure matters and should count in the paper's favor.

The main soft spot is exactly that dependency. Proposition 4.1, Lemma 4.2, and Proposition 5.2 are load-bearing: the inverse and product moments feed every later comparison, and if [Ope26b] has a normalization error, the constant has no support. The numerical margin over Newton's fixed-field barrier is only about 2.6e-9, so even a small error matters. A second soft spot is that Sections 13 through 20 were not in the text I reviewed, so Theorem 14.11 and the Section 18 descent to every number field are asserted rather than visible. That is a verification limit, not a detected flaw.

One concern from the report I would push back on: the fixed-point discharge in Remark 3.1 is not circular in a harmful sense. In the contradiction argument, assuming beta* > Bflat and setting kappa = 2 beta* - 1 makes the moment hypothesis beta* <= (1+kappa)/2 true by definition of the supremum. The real question is whether the product moment stays uniform down to kappa_flat, and that is a missing-sections question.

This paper is for specialists in moment methods and explicit zero-free regions. It deserves a serious referee, with the condition that [Ope26b] be posted and the full Part II/III be made available. I would not cite it until then.

What would settle it

The cheapest check is arithmetic: evaluate $133488B^4 - 376596B^3 + 384534B^2 - 167541B + 26231$ at $B = 0.874886647039964$ and at $B = 0.874886647039965$ and confirm the sign change that locates the stated root. The decisive experiment is a numerical zero search: compute zeros of finite-order Hecke $L$-functions over, say, $\mathbb{Q}(\zeta_{12})$ or a class-field extension of it, and look for any zero with real part strictly between $0.874886647039964$ and $1$ — the theorem forbids such a zero. A third check isolates the fragile import: recompute, directly from the Jacquet modules of the exceptional representation, the local Whittaker recurrence $J_{e+3} = q^{-1}J_e$ and the Gauss identity $\gamma_2(p)^3 = -\Lambda(p)$ of Lemma 4.2, independently of the preprint.

Referee report

Significance

If the analytic inputs are correct, this is a genuine uniformity result: it improves the known uniform zero-free half-plane constant by roughly 7.0e-5 over B_new and 2.6e-9 over Newton's boundary, and it removes the restriction to cyclotomic or abelian base fields. The scalar layer is unusually explicit: the quartic is stated, Appendix B gives an exact rational chain, the Bernstein-basis inequalities of Section 2 are displayed with bounds, and the fixed-point contradiction is internally coherent. I do not see circularity in the limiting argument: setting kappa=2*beta*(F)-1 makes the M2 hypothesis (1+kappa)/2=beta*(F) an equality, which is an admissible discharge of that hypothesis rather than a reuse of the conclusion being proved. The importance of the paper is nevertheless conditional on the correctness and normalizations of the imported exceptional-theta construction, since the whole moment chain of Sections 10--12 rests on it and the numerical improvement over [New26] is very small.

Major objections (2)

[Sections 4--5, Remark 3.1; Theorems 10.4, 11.2, 12.14] The central analytic engine is not self-contained. Proposition 4.1 imports the finite small-norm bound from [Ope26b, Section 5.2, (5.13)], Lemma 4.2 imports [Ope26b, Lemma 3.4, equations (3.9)--(3.14)], and Proposition 5.2's radial functional equation and uniform bounds are stated as consequences of [Ope26b, Section 5]. All three moment theorems invoke Propositions 4.1 and 5.2, and the comparison chain of Sections 13--14 depends on those moments; without [Ope26b] the boundary B_flat has no independent support. The abstract and Remark 3.1 disclose this dependency, but the main theorem is stated unconditionally. Because the gain over [New26] is only about 2.6e-9, even a small normalization discrepancy in the local Whittaker table or in the radial bounds could change the sign of the comparison. Please either include the needed statements and proofs of the exceptional-theta construction, or state the theorem as conditional on [Ope26b], and verify explicitly that the normalizations of [Ope26b] match every use made in Sections 10--12.
[Section 18 and Theorem 17.1] The descent to an arbitrary number field K rests on the Section 18 assertion that the entire Part II argument uses only mu_12 subset F and standard facts valid for every number field. The manuscript states this as a check and lists standard inputs (S-unit and class-group finiteness, Hilbert reciprocity, Kummer theory, Kazhdan--Patterson theory over fields containing mu_3, and the large sieves), but it does not demonstrate how the Proposition 4.1 local Whittaker table, the Lemma 4.2 coefficient-character construction, and the Proposition 5.2 radial bounds extend from the abelian cyclotomic fields F=Q(zeta_m) fixed in Section 3 to F=K(zeta_12) for a general non-abelian K. This is load-bearing: Theorem 3.2 covers only the class-field pairs (K,eta) in the set C, while the Main Theorem claims every number field. Please supply the field-uniformity verification, or restrict Theorem 17.1 to the statement supported by the displayed argument.

Minor objections (4)

[Abstract and Introduction] The formula for B_new is typeset ambiguously as '18xi_new + 718xi_new + 9'; it should be displayed as (18xi+7)/(18xi+9) to match (2.1) and the surrounding text.
[Title and table of contents] The running header and table of contents read 'HECKEL-FUNCTIONS'; this should be 'HECKE L-FUNCTIONS'.
[Throughout] The symbol B is heavily overloaded: it denotes the numerical boundary B_flat or B_new, the detected element set B in Section 2.2 and Proposition 2.2, and a norm bound B in Lemma 7.1. Renaming at least the detected set or the lemma bound would substantially improve readability.
[Remark 3.1] The distinction between the fully formalized scalar/arithmetic parts and the not-yet-formalized analytic part of Part II is clearly stated; I would suggest also adding a single sentence in the abstract noting that the analytic moment estimates themselves are not part of the Lean formalization, to avoid overstating the machine-checked scope.

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

The paper

The paper claims

For every number field $K$ and every finite-order Hecke character $\chi$, the Hecke $L$-function $L_K(s,\chi)$ is zero-free in $\operatorname{Re}(s) > B_{\flat} = 0.874886647039964\ldots$, with the principal character's pole at $s=1$…

The reading

The paper claims a uniform quasi-Riemann hypothesis: the single constant $B_{\flat} = 0.874886647039964\ldots$ — the root near $0.8749$ of the explicit quartic $133488B^4 - 376596B^3 + 384534B^2 - 167541B + 26231 = 0$ — is a zero-free boundary for every finite-order Hecke $L$-function over every number field, so in particular every Dedekind zeta function $\zeta_K(s)$ has no zeros (other than the permitted pole at $s=1$) in $\operatorname{Re}(s) > B_{\flat}$, regardless of $K$ or the character's conductor. This improves the previous uniform bound of $31/32$ for arbitrary number fields and the recent $0.874957\ldots$ bound for $\mathbb{Q}(\sqrt{-3})$, and the paper claims $B_{\flat}$ is the exact infimum of its refined comparison system: no finite chain of admissible steps reaches below it. The proof extends the cubic-$\theta$ moment argument to cyclotomic fields containing the $12$th roots of unity, replacing the cubic $\theta$ function by the exceptional $\theta$ representation of Kazhdan–Patterson, and then descends to every number field by running the argument over $K(\zeta_{12})$ and using class-field factorization. The result matters because it moves the unconditional zero-free region for an entire family of global $L$-functions closer to the critical line, uniformly in conductor, at an explicitly stated numerical constant.

What carries the argument

The engine is the exceptional $\theta$ representation of Kazhdan–Patterson — the automorphic representation attached to the rank-two, degree-three metaplectic cover, normalized so that its local Whittaker table has $W_p(a(\varpi^e u)) = \chi_p(u)^{2e}W_p(a(\varpi^e))$, nonzero exactly for $e \equiv 0, 1 \pmod 3$ with values $q_p^{-k}$ and $q_p^{-k-1/2}\gamma_2(p)$, and obeying the recurrence $J_{e+3} = q^{-1}J_e$. Over a cyclotomic field $F \supset \mu_{12}$ it replaces the cubic $\theta$ function of the $\mathbb{Q}(\sqrt{-3})$ argument; its radial functional equation (Proposition 5.2) turns reflected norm-sheet sums into compactly supported smooth tests and thereby carries the inverse moments (Theorems 10.4 and 11.2) and the unweighted product moment for $13/18 \leq \kappa \leq 1$ (Theorem 12.14). The scalar comparison system is the second engine: the endpoint exponents $E_{\mathrm{pt}}$ (pointwise numerator) and $E_{\mathrm{av}}$ (averaged numerator), the tangent-parameter family of Lemma 14.10, and the fixed-point identity $F_{\flat}(\kappa_{\flat}) = (1+\kappa_{\flat})/2 = B_{\flat}$ with $\kappa_{\flat} = 2B_{\flat} - 1$, which becomes the quartic defining $B_{\flat}$. Class-field descent, using only $\mu_{12} \subseteq F$ and general-field sieve and reciprocity inputs, then lifts the cyclotomic statement to every number field.

Extended reading notes

Core claim

The paper's central claim, the Main Theorem (Theorem 17.1; Theorem 3.2 for cyclotomic base fields and their class-field descendants), states that for every number field $K$ and every finite-order Hecke character $\chi$ of $K$, $L_K(s,\chi) \neq 0$ for $\operatorname{Re}(s) > B_{\flat}$, where $B_{\flat} = 0.874886647039964\ldots$ is the root near $0.8749$ of $133488B^4 - 376596B^3 + 384534B^2 - 167541B + 26231 = 0$; the pole of the principal character at $s=1$ is permitted, and in particular $\zeta_K(s) \neq 0$ for $\operatorname{Re}(s) > B_{\flat}$, $s \neq 1$, with no assertion on the boundary line itself. The paper further claims that $B_{\flat}$ is the exact infimum of the refined comparison system built from the moment parameter $\kappa$ and the averaged-numerator estimate: every finite chain of admissible refined steps starting from a bound at least $B_{\flat}$ produces only boundaries at least $B_{\flat}$, yet chains starting from $7/8$ approach $B_{\flat}$ from above within any $\varepsilon$, so the limiting contradiction argument attains an endpoint no finite chain can reach. The larger boundary $B_{\infty} = B_{\mathrm{new}} = 0.874957019420098946\ldots$ is separately shown to be the infimum of the pointwise-numerator system, and the improvement to $B_{\flat}$ is attributed concretely to replacing the pointwise reflected-numerator bound by an amplified first moment. Because the analytic argument is asserted to use only the containment $\mu_{12} \subseteq F$ plus standard facts valid for every number field, the cyclotomic theorem is run over $F = K(\zeta_{12})$ and descended to arbitrary $K$ by class-field factorization; the paper stresses that the $\theta$ constructions and local calculations are imported from the preprint [Ope26b] but not that preprint's zero-free conclusion, and it states in Remark 3.1 that the accompanying machine-checked formalization covers only scalar and selected arithmetic implications, not the complete analytic argument of Part II.

Load-bearing premise

The engine of the argument — the exceptional $\theta$ representation over a cyclotomic field containing the $12$th roots of unity, with its local Whittaker table, radial functional equation, and small-norm bounds — is imported from the unpublished preprint [Ope26b], and the abstract acknowledges that the moments rest on it; if that construction or its normalizations are wrong, the boundary $B_{\flat}$ has no support, and likewise if the Section 18 claim that only $\mu_{12} \subseteq F$ and standard facts are used should fail for some number field, the descent to every $K$ collapses.

Editorial extensions

If this is right

A uniform zero-free half-plane at $B_{\flat} = 0.874886647039964\ldots$ holds for $\zeta_K$ and for every finite-order Hecke $L$-function over every number field, uniformly in the field, the character, and the conductor, with the principal pole at $s=1$ the only permitted singularity.
The boundary improves the two benchmarks cited: the uniform $31/32$ region for arbitrary number fields from [CT26] and the recent $B_{\mathrm{new}} \approx 0.874957$ for $\mathbb{Q}(\sqrt{-3})$; over $\mathbb{Q}(\sqrt{-3})$ the boundary $8748867/10^7$ of the averaged-numerator method is recovered as a stage in the rational chain of Appendix B.
Optimality within the method: $B_{\flat}$ is the exact infimum of the refined comparison system — every finite chain of admissible refined steps starting above $B_{\flat}$ yields only boundaries $\geq B_{\flat}$, while chains from $7/8$ come within any $\varepsilon$ of $B_{\flat}$ from above — so the bound is a sharp output of the stated system, reached only by the limiting contradiction argument.
Arithmetic scope: the cyclotomic theorem descends to every pair $(K,\eta)$ whose cyclic class field becomes abelian over a cyclotomic field containing $\mu_{12}$ (in particular every finite-order character over every abelian number field), and the $\mu_{12}$-only argument extends the boundary to every number field.

Reading between the lines

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

A consequence the paper leaves implicit: because parameter variation inside the refined comparison system cannot go below $B_{\flat}$, any future reduction of the boundary must come from one of the inputs the paper itself flags as capable of changing the obstruction — a stronger mean-value estimate, a joint moment for the detecting polynomials, larger admissible prime lengths, or better functional
The descent template is portable: the same scheme — prove a moment estimate over $K(\zeta_{12})$ using only the root-of-unity condition, then push down by class-field factorization — should apply to neighbouring families, such as norm-twisted or higher-degree automorphic $L$-functions, whenever an exceptional theta input exists over a cyclotomic field; Part III's analysis shows how little of the a
An exact-arithmetic foothold the paper supplies: the rational chain of Appendix B ending at $B_{\sharp} = 874886647041/10^{12}$, within $1.04 \times 10^{-12}$ of $B_{\flat}$, is independently checkable and would expose any slip in the quartic or its root even without the analytic argument.

Original abstract

In [Liu26a], varying the length and moment parameters in the cubic-theta argument of [Ope26a] gave the zero-free half-plane $Re(s)>B_{\mathrm{new}}=0.874957019420098946\ldots$ for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and for Dirichlet $L$-functions, and its optimality in the specified exponent comparison system. In this paper, we extend the argument to finite-order Hecke $L$-functions over cyclotomic fields containing $\mu_{12}$ and, by class-field descent, to every pair $(K,\eta)$ whose cyclic class field has metabelian normal closure over $\mathbb Q$, including every finite-order character over every abelian number field. Then, adapting the averaged numerator estimate developed by Xu and Sang [XS26] for Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and combining it with a limiting contradiction argument, we obtain the boundary $B_\flat=0.874886647039964\ldots$, the root near $0.8749$ of $133488B^4-376596B^3+384534B^2-167541B+26231$; it is the exact infimum of the refined comparison system, unattained by finite chains starting above it. Since the analytic argument uses only $\mu_{12}\subset F$, we run it over $F=K(\zeta_{12})$ for an arbitrary number field $K$ and descend: every finite-order Hecke $L$-function over every number field is zero-free for $Re(s)>B_\flat$, giving a uniform Quasi-Riemann Hypothesis over number fields. The analytic argument uses the exceptional-theta constructions and local calculations of the preprint [Ope26b].

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, 2026-10-20: 1 request, 5,007 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.30 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.93 · 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.

In [Liu26a], varying the length and moment parameters in the cubic-theta argument of [Ope26a] gave the zero-free half-plane $Re(s)>B_{\mathrm{new}}=0.874957019420098946\ldots$ for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and for Dirichlet $L$-functions, and its optimality in the specified exponent comparison system. In this paper, we extend the argument to finite-order Hecke $L$-functions over cyclotomic fields containing $\mu_{12}$ and, by class-field descent, to every pair $(K,\eta)$ whose cyclic class field has metabelian normal closure over $\mathbb Q$, including every finite-order character over every abelian number field. Then, adapting the averaged numerator estimate developed by Xu and Sang [XS26] for Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and combining it with a limiting contradiction argument, we obtain the boundary $B_\flat=0.874886647039964\ldots$, the root near $0.8749$ of $133488B^4-376596B^3+384534B^2-167541B+26231$; it is the exact infimum of the refined comparison system, unattained by finite chains starting above it. Since the analytic argument uses only $\mu_{12}\subset F$, we run it over $F=K(\zeta_{12})$ for an arbitrary number field $K$ and descend: every finite-order Hecke $L$-function over every number field is zero-free for $Re(s)>B_\flat$. The analytic argument uses the exceptional-theta constructions and local calculations of the preprint [Ope26b].

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	In [Liu26a], varying the length and moment parameters in the cubic-theta argument of [Ope26a] gave the zero-free half-plane $Re(s)>B_{\mathrm{new}}=0.874957019420098946\ldots$ for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and for Dirichlet $L$-functions, and its optimality in the specified exponent comparison system.	background 0.99	0.16	0.05	asserted 1.00
s1	In this paper, we extend the argument to finite-order Hecke $L$-functions over cyclotomic fields containing $\mu_{12}$ and, by class-field descent, to every pair $(K,\eta)$ whose cyclic class field has metabelian normal closure over $\mathbb Q$, including every finite-order character over every abelian number field.	contribution 0.98	0.84	0.17	asserted 1.00
s2	Then, adapting the averaged numerator estimate developed by Xu and Sang [XS26] for Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and combining it with a limiting contradiction argument, we obtain the boundary $B_\flat=0.874886647039964\ldots$, the root near $0.8749$ of $133488B^4-376596B^3+384534B^2-167541B+26231$; it is the exact infimum of the refined comparison system, unattained by finite chains starting above it.	result 0.95	0.93	0.30	asserted 1.00
s3	Since the analytic argument uses only $\mu_{12}\subset F$, we run it over $F=K(\zeta_{12})$ for an arbitrary number field $K$ and descend: every finite-order Hecke $L$-function over every number field is zero-free for $Re(s)>B_\flat$.	result 0.52	0.85	0.81	asserted 1.00
s4	The analytic argument uses the exceptional-theta constructions and local calculations of the preprint [Ope26b].	method 0.83	0.15	0.02	asserted 0.99

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

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.