New zero-free regions for Dedekind zeta-functions at small and large ordinates

Aditi Savalia; Ethan Simpson Lee; Peng-Jie Wong; Sourabhashis Das; Swati Gaba

REVIEW 4 major objections 5 minor 1 cited by

New zero-free regions for Dedekind zeta-functions at small and large ordinates

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper establishes new explicit zero-free regions for Dedekind zeta-functions, with sharper large-ordinate constants and low-ordinate results that now cover every number field of degree at least 2.

desk verdict Careful, incremental improvement to explicit zero-free regions for Dedekind zeta-functions; the small-ordinate extension to all number fields is the real news, and the weaknesses are transparency, not math. read the letter →

arxiv 2506.19319 v1 pith:NTRSK4KB submitted 2025-06-24 math.NT

Sourabhashis Das , Swati Gaba , Ethan Simpson Lee , Aditi Savalia , Peng-Jie Wong This is my paper · ORCID

classification math.NT MSC 11R42

keywords Dedekindzeta-functionszero-freeregionsexplicitformulaetrigonometricpolynomialsexceptionalzerosChebotarevdensitytheoremgammafactorsnumberfields

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper seeks to pin down where the nontrivial zeros of Dedekind zeta-functions cannot live: it produces explicit zero-free regions with better constants than all previous published regions. For number fields of degree at least 3, the main large-ordinate result (Theorem 1.1) gives a region of the form $\sigma \geq 1 - 1/(C_1 \log d_L + C_2 n_L \log|t| + C_3 n_L + C_4)$ valid for $|t| \geq 1$, with the best constants to date. For low-lying zeros, the paper removes a 'sufficiently large discriminant' restriction and gives degree-by-degree constants valid for every number field of degree at least 2, allowing at most one exceptional real zero. These improvements matter because zero-free regions directly control the error term in effective versions of the Chebotarev density theorem and in related prime-ideal problems.

What carries the argument

The engine is the non-negative trigonometric-polynomial method. One chooses a polynomial $p_n(\varphi)=\sum_{k=0}^n a_k \cos(k\varphi)$ with $a_k \geq 0$, $a_0<a_1$, and $p_n(\varphi) \geq 0$ for all real $\varphi$, then forms $S(\sigma,t)=\sum_{k=0}^n a_k f_L(\sigma,kt)$, where $f_L$ is built from the logarithmic derivative of $\zeta_L$ evaluated at a shifted point $s'_k$; the shift and the weight $\kappa=1/\sqrt{5}$ are chosen so that positivity of $p_n$ forces $S(\sigma,t)\geq 0$. An explicit formula then bounds $S$ above by contributions from zeros, the pole, the discriminant, and the gamma factors. The two load-bearing refinements are a sharper bound for the gamma-factor contribution (Lemma 2.1) and the use of high-degree candidate polynomials $p_{40}$ and $p_{46}$ alongside earlier choices $p_8$ and $p_{16}$.

What would settle it

Take the coefficient tables for $p_{40}$ and $p_{46}$, evaluate the polynomials at a fine grid of angles from 0 to $\pi$ with high-precision arithmetic, and check both the minimum value and the sign of $a_1-a_0$. A single negative value, or $a_0 \geq a_1$, would invalidate the inequality $S(\sigma,t)\geq 0$ on which the zero-free region is built; an exact sum-of-squares certificate would confirm the needed property.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a set of numerically optimized explicit exclusion regions for Dedekind zeta-functions. The central claim is that if $L$ has degree $n_L \geq 3$, then $\zeta_L(\sigma+it) \neq 0$ for $|t| \geq 1$ whenever $\sigma$ lies above the curve defined by $(C_1,C_2,C_3,C_4)=(12.21124,\,9.54177,\,-11.59548,\,4.57803)$, and that the same style of computation yields admissible constants for every degree $n_0$ between 3 and 21. For $0<|t|\leq 1$, the paper claims at most one exceptional zero in the region $\sigma \geq 1 - 1/(A \log d_L)$ with $A'=0$ and degree-dependent $A$ values, extending an earlier result to all number fields. When a real exceptional zero is present, still larger zero-free regions are claimed, with constants depending on how close that zero lies to 1; the real-line case similarly allows at most one real zero with explicit constants for every degree.

Load-bearing premise

Everything downstream rests on the trigonometric polynomials $p_{40}$ and $p_{46}$ being genuinely non-negative on the whole real line and satisfying $a_0<a_1$; the paper relies on coefficient tables for these polynomials without re-proving that property, and the key inequality $S(\sigma,t)\geq 0$ collapses if it fails.

Editorial extensions

If this is right

Any effective application that uses zero-free regions for Dedekind zeta-functions, in particular the error term in the Chebotarev density theorem, directly inherits the improved constants.
For every number field of degree $n_L \geq 3$, Theorem 1.1 supplies a zero-free region for all $|t| \geq 1$ with fixed constants and no case split by discriminant size.
The low-ordinate theorem gives at most one exceptional zero in the region (1.3) for every degree $n_L \geq 2$ with $A'=0$, removing the 'sufficiently large discriminant' condition from earlier work.
If an exceptional zero is known to lie in $[1-\nu/\log d_L, 1)$, the enlarged regions of Theorem 1.3 apply; the $\nu=0.05$ case matches the assumption used in the least-prime-ideal problem.
On the real line, at most one real zero is allowed in region (1.4) for every degree, with explicit $A''$ values for all $n_L \geq 2$.

Reading between the lines

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

The same optimization could be rerun with the non-negativity of the trigonometric polynomials certified by exact rational or sum-of-squares proofs, which would make the constants independently reproducible and open the door to automated searches over even higher degrees.
The large negative constant $C_3$ for small degrees suggests this particular method is close to its ceiling for $n_L$ around 3 to 5; further progress may need sharper zero-repulsion or large-sieve input rather than only better polynomials.
Because the low-ordinate constants now hold uniformly for all number fields of fixed degree, effective Chebotarev bounds can be stated without a discriminant threshold; a testable consequence is that least-prime-ideal bounds would improve by the ratio of the old to the new constants.
The trade-off between $p_{16}$ (best $C_2$) and $p_{46}$ (best $C_1$) indicates there is no single optimal polynomial; a two-parameter family balancing the $\log|t|$ and $\log d_L$ terms could yield a Pareto front of zero-free regions rather than one fixed set of constants.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this is a genuine improvement on Lee's explicit zero-free regions for Dedekind zeta-functions, and the small-ordinate result (Theorem 1.2) removes Kadiri's 'sufficiently large discriminant' condition entirely—that is the most substantive new result. Second, the proof is a well-executed run of the Stečkin/Kadiri machinery, not a new method; its value is in the constants and in the removal of the discriminant restriction.

The genuinely new pieces: Lemma 2.1, a refined gamma-factor uniform bound, with a complete proof; and the use of Mossinghoff–Trudgian–Yang's p40/p46 polynomials in this setting. The authors also correct two typos in Kadiri's paper, which is a good sign of careful reading. The main derivation is standard: S(σ,t) ≥ 0 via nonnegative trigonometric polynomial, explicit formula, case split. I did not find a circular step; the optimization parameters are chosen to minimize the final constants, which is normal for this genre.

The soft spots, in order of severity. First, the nonnegativity of p40 and p46 is imported from [15] without any check or certificate. That is load-bearing: S(σ,t) ≥ 0 fails if the polynomial dips negative. In practice these are published, well-tested polynomials, and citing them is standard; but a referee should ask the authors to state explicitly that nonnegativity is verified in [15] and, ideally, to include a quick verification script. Second, the numerical optimization is opaque: 'we have determined' with no code, data, or detailed algorithm for the searches. That's common in this literature but annoying; shipping a script would make the paper fully reproducible. Third, the structure paragraph misstates that Theorem 1.1 is proved in Sections 4–5; it's proved in Section 3. Cosmetic, but sloppy in a paper that otherwise pays close attention to detail. I'd also note Lemma 2.2's 'optimising these values' is a bit hand-wavy, though the table is there.

Bottom line: this deserves a serious referee. The central argument holds up as far as I can tell; the concerns are transparency, not correctness. I'd send it out and ask for the computational verification and a fix of the structure paragraph. If the authors supply code or at least a detailed description of the optimization, I'd be happy to see it accepted.

Referee Report

4 major / 5 minor

Summary. The paper develops explicit zero-free regions for Dedekind zeta-functions of number fields L≠Q. Theorem 1.1 gives constants (C1,C2,C3,C4,T)=(12.21124,9.54177,-11.59548,4.57803,1) for the region (1.2) valid for degree nL≥3, improving Lee's earlier large-ordinate constants. Theorems 1.2–1.5 treat zeros with |γ|≤1 and real zeros, extending Kadiri's low-lying results to all number fields, removing the 'sufficiently large' condition via small-discriminant GRH verifications, and giving Deuring–Heilbronn-type refinements under an exceptional zero. The proof follows the Stečkin–Kadiri–Lee method, with a new gamma-factor estimate (Lemma 2.1) and choices of trigonometric polynomials from Mossinghoff–Trudgian–Yang. The argument reduces the existence of a zero to bounding the four terms S1–S4 in the explicit formula; Case 1 (|γ|>1) yields Theorem 1.1, and Cases 2–5 yield the remaining theorems.

Significance. If the constants withstand independent verification, Theorem 1.1 gives the current best explicit zero-free region for Dedekind zeta-functions at large ordinates, and Theorems 1.2–1.5 extend Kadiri's low-lying results to all number fields while improving several constants. The paper corrects two typographical errors in Kadiri's work, provides a complete proof of Lemma 2.1, and carefully tracks the optimization parameters. The main weakness is that the headline numerical results rely on imported nonnegativity of the polynomials p40 and p46 and on optimization computations for which no code or data are supplied; these need to be certified before the results can be taken as established. This is a verification gap rather than a defect in the underlying analytic argument.

major comments (4)

[Section 3.3 and Tables 12–13] The proof of Theorem 1.1 makes essential use of the assumption p_n∈P_n, i.e., p_n(φ)≥0 for all φ and a0<a1, for n=40 and n=46. These polynomials are imported from [15], but the manuscript only lists their coefficients in Tables 12–13 and does not reproduce or reference a verification of nonnegativity. Since [15] is an arXiv preprint (arXiv:2212.06867), this is not a settled published input. The inequality S(σ,t)≥0 in Section 2, Eq. (2.3), and hence the lower bound in Case 1, collapses if p_n fails to be nonnegative. Please add a certificate (e.g., a Fejér–Riesz factorization or a rigorous numerical proof) for p40 and p46, or supply a complete published reference in which these properties are proved.
[Section 4.2, Eq. (4.18)] The paper asserts d_L≥400001 for nL=2, attributing this to Tollis's verification work [19]. However, the version of Tollis's result quoted in the introduction covers only nL=3 with d_L≤239 and |t|≤92, and nL=4 with d_L≤320 and |t|≤40; it says nothing about quadratic fields. Thus the nL=2 rows of Tables 3, 6, and 7 and the nL=2 statements of Theorems 1.2 and 1.4 are not justified by the cited source. If a GRH verification for quadratic fields of conductor <400001 exists (for example, from Platt's work on Dirichlet L-functions), it must be cited explicitly; otherwise the lower bound (4.18) for nL=2 should be removed or proven.
[Lemmas 3.1 and 3.2, Section 3.1] These lemmas provide the bounds on S3 and S4 that feed directly into the final inequality (3.2) and hence Theorem 1.1. Their proofs are not given; the reader is referred to 'arguments in [12, Sec. 2.2]' and '[12, Sec. 2.3]'. Since Lemma 3.2 introduces a new piecewise definition (C1(k,δ,ε), C2(k,δ,ε), A(k,t,δ,ε)) and Lemma 2.1 changes the gamma-factor bound, the adaptation is not a routine citation. Please include complete proofs or a detailed derivation of these two lemmas.
[Section 3.4 and Tables 2–7, 9–11] The optimized constants and parameter choices are presented without a reproducible audit trail: 'straightforward optimisation methods', 'we have determined', and the algorithm in Section 3.4 with step size 0.0001 do not allow the reader to verify the quoted 5-6 digit constants. Since the contribution of the paper is precisely these numerical constants, please provide the computer code or detailed pseudo-code with sufficient precision, or include an extended derivation of the entries in Tables 2–7 and 9–11.

minor comments (5)

[Section 1, Structure paragraph] The sentence 'We will prove Theorem 1.1 in Sections 4 (for complex zeros) and 5 (for real zeros)' conflicts with the preceding sentence and with the actual content: Theorem 1.1 is proved in Section 3, while Sections 4–5 prove Theorems 1.2–1.5.
[Section 3.2] The displayed formula for the maximizing r reads 'r=√a0√a1−√a0'; it should be r=√a0/(√a1−√a0).
[Section 3.4] The iterative loop uses ε increment 0.0001; constants in Tables 9–11 are quoted to 5 decimals, so the last digits of C4 should be reported with rounding to the precision of the search, or the optimization should be rerun at higher precision.
[References] [15] is an arXiv preprint (arXiv:2212.06867); a published version should be cited if available.
[Lemma 2.2 and Table 8] The phrase 'Optimising these values' preceding Table 8 gives no indication of the method; a short description of the optimization (e.g., checking the stationary points of the explicit expression) would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No substantive circularity: the new constants are produced by a standard explicit-formula argument with external trigonometric polynomials; no step assumes the conclusion being proved.

full rationale

The derivation chain is self-contained in the relevant sense: the target zero-free regions are deduced from the explicit formula (2.2), auxiliary bounds for zero sums (3.1), Lemmas 2.1, 3.1, 3.2, and algebraic rearrangement culminating in (3.2)–(3.3). The constants (C1,C2,C3,C4) are not fitted to the zeros of ζL or to the statement being proved; they are computed from the coefficients of chosen trigonometric polynomials and from independent bounds. The imported polynomials p40 and p46 from Mossinghoff–Trudgian–Yang [15] are external, parameter-free objects whose defining property p_n(φ)≥0 does not involve Dedekind zeta-functions or the constants in question, so relying on their published coefficients and nonnegativity is a normal use of a citation, not a circular reduction. The self-cited works ([6], [8], [12]) supply auxiliary lemmas, numerical groundwork, and an earlier constant set; notably [12] is an independent published result by the third-named author, and the present paper's improvement over it is exactly the claimed new content. The absence of a reproduced nonnegativity certificate for p46 is a verification/correctness risk, not circularity, because the property is checkable independently of the paper's conclusions. No equation in the paper defines a predicted quantity in terms of itself, and no fitted parameter is renamed as a prediction. The numerical optimization of ε, r, d1, d2, and the polynomial choice is standard parameter selection for explicit bounds and does not force the theorem's truth by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard analytic number theory facts plus a few external inputs: the non-negativity of tabulated polynomials, the discriminant lower bounds from Kadiri-Wong, and small-case GRH verifications. All optimization parameters (r, d1, d2, epsilon) are chosen to minimize the final constants; they are part of the proof method, not fitted to the target result. No new mathematical entities are introduced.

free parameters (5)

epsilon ε for large-ordinate constants (Theorem 1.1) = 0.1239 for n0=3 with p46; decreases to 0.0030 for n0=21
Chosen as the least value (to 4 decimal places) satisfying condition (3.4) in Section 3.4; it affects C3 and C4 in (3.3).
r, d2 for Case 2 (Table 2) = e.g., nL=2: r=0.201, d2=0.70445; nL=3: r=0.165, d2=0.56927
Optimization variables in τ1(r,d2) that determine A for 0 < |γ0| ≤ 1 in Theorem 1.2; chosen to minimize τ1^{-1}.
rA, rB, d1, d2 for Cases 3 and 4 (Table 3) = e.g., nL=3: rA=0.3348, rB=0.8220, d1=0.4140, d2=0.56927
Optimization variables balancing τ2 and τ3 for Theorem 1.2; d2 is fixed from Table 2.
r, d1, d2 for the exceptional-zero case (Tables 4 and 5) = e.g., ν=0.05, nL=2: rA=0.54, rB=0.58, rC=1.93, d1=2.14, d2=2.14
Optimization variables for Theorems 1.3 and 1.5 under an assumed exceptional zero in [1 - ν/L, 1).
r for the real-zero case (Table 6) = e.g., nL=2: r=1.49859; nL=3: r=0.822093
Optimization parameter in η(nL,r) for Theorem 1.4, with a discretization step of 0.00001 in the algorithm below (5.4).

assumptions (6)

standard math The explicit formula for the logarithmic derivative of ζ_L (equation 2.2)
Quoted from Lagarias-Odlyzko [11, Eqn. (8.3)] and used throughout to relate zeros to log dL and gamma factors.
domain assumption The chosen trigonometric polynomials p_n are non-negative and have a0 < a1
Non-negativity is the defining property of P_n in Section 2 and is what makes S(σ,t) ≥ 0. The paper assumes p40 and p46 from [15] satisfy this without reproducing a verification.
domain assumption Discriminant lower bounds dL ≥ dmin(nL) from [8, Appendix]
Imported without proof in Section 2.1, Table 1, and used in the optimization for each degree.
domain assumption Tollis' GRH verification and Kadiri's Dirichlet L-function zero-free region for nL=2 (inequality 4.18)
Section 4.2 assumes dL ≥ 400001 for nL=2, 240 for nL=3, and 321 for nL=4, based on external numerical results.
standard math Kadiri's Lemmas 4.1-4.3 and the bounds (4.2)-(4.9)
Cited from [6] and used to bound the zero sum S1. The paper corrects two typos in [6] but otherwise relies on these published results.
standard math Standard analytic continuation and functional equation of ζ_L
Used implicitly in the completed zeta function and in the explicit formula.

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Pith. "Pith review of New zero-free regions for Dedekind zeta-functions at small and large ordinates." pith.science (2026). https://pith.science/paper/NTRSK4KB

@misc{pith2026250619319,
  author       = {Pith},
  title        = {Pith review of: New zero-free regions for Dedekind zeta-functions at small and large ordinates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NTRSK4KB}},
  note         = {Machine review of arXiv:2506.19319}
}

abstract

Given a number field $L\neq \mathbb{Q}$, we obtain new and explicit zero-free regions for Dedekind zeta-functions of $L$, which refine the previous works of Ahn--Kwon, Kadiri, and Lee. In particular, for low-lying zeros, we extend Kadiri's result to all number fields while improving the main constant.

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Cited by 1 Pith paper

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

An effective version of Chebotarev's density theorem
math.NT 2025-08 unverdicted novelty 6.0 of 10

A fully explicit refinement of Lagarias and Odlyzko's effective Chebotarev theorem for all non-rational number fields, with a sharper error term for small-degree extensions.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [15]

M. J. Mossinghoff, T. S. Trudgian, and A. Yang, Explicit zero-free regions for the Riemann zeta-function , arXiv:2212.06867 (2022)

work page Pith review arXiv 2022

[2] [19]

Tollis, Zeros of Dedekind zeta functions in the critical strip , Math

E. Tollis, Zeros of Dedekind zeta functions in the critical strip , Math. Comp. 66 (1997), no. 219, 1295–1321. MR1423079

work page 1997

[3] [1]

Ahn and S.-H

J.-H. Ahn and S.-H. Kwon, An explicit upper bound for the least prime ideal in the Chebotarev density theorem, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 3, 1411–1458. MR3986919

work page 2019

[4] [2]

Das, An explicit version of Chebotarev’s density theorem , MSc Thesis, University of Lethbridge, 2020

S. Das, An explicit version of Chebotarev’s density theorem , MSc Thesis, University of Lethbridge, 2020

work page 2020

[5] [3]

C. J. de la Vall´ ee Poussin,Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inf´ erieurs ` a une limite donn´ ee. M´ em. Acad. r, M´ em. Couronn´ es et Autres M´ em. Publ. Acad. Roy. Sci. Lett. Beaux-Arts Belg. 59 (1899), 74

work page

[6] [4]

Kadiri, Une r´ egion explicite sans z´ ero pour les fonctions l de dirichlet , PhD Thesis, Universit´ e Lille I, 2002

H. Kadiri, Une r´ egion explicite sans z´ ero pour les fonctions l de dirichlet , PhD Thesis, Universit´ e Lille I, 2002

work page 2002

[7] [5]

117 (2005), no

, Une r´ egion explicite sans z´ eros pour la fonctionζ de Riemann, Acta Arith. 117 (2005), no. 4, 303–

work page 2005

[8] [6]

, Explicit zero-free regions for Dedekind zeta functions , Int. J. Number Theory 8 (2012), no. 1, 125–

work page 2012