Estimating the number of zeros of Dedekind zeta-functions

Victor Amberger

REVIEW 2 major objections 4 minor 1 cited by

Estimating the number of zeros of Dedekind zeta-functions

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A new explicit bound improves zero-counting for Dedekind zeta functions, including the Riemann zeta case.

desk verdict Genuinely new operator method for zero-counting errors, with improved constants, but the key lemma's numerical verification is not rigorous and needs a certified proof before the theorem can be accepted. read the letter →

arxiv 2510.27444 v4 pith:NWA32TS7 submitted 2025-10-31 math.NT

Victor Amberger This is my paper · ORCID

classification math.NT MSC 11M2611M0611R42

keywords DedekindzetafunctionzerocountingexpliciterrorestimatesRiemannargumentprinciplenontrivialzerosnumberfieldsharmonicinequalities

topics The Riemann Hypothesis

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 sets out to improve the explicit error term in the asymptotic formula for N_K(T), the number of nontrivial zeros of a Dedekind zeta function ζ_K with imaginary part at most T counted with multiplicity and boundary half-weight. The proof applies the argument principle to a shifted rectangle and bounds the resulting sum over zeros using a harmonic inequality with four numerically optimized parameters. For every number field K it produces a uniform error term valid for all T ≥ 1, and specializing to K = Q gives a new explicit bound for the Riemann zeta function. If the argument is correct, these are the sharpest explicit zero-counting error terms currently available.

What carries the argument

The engine of the proof is Lemma 2.1, which bounds the arctangent sum f(b,t,d) = 2 arctan((b+d)/t) + 2 arctan((−b+d)/t) − arctan((b+2d)/t) − arctan((−b+2d)/t) between two rational functions of t with parameters d, a1, a2, a3. The lemma asserts a harmonic function H(b,t) is never positive on |b| ≤ 1/2, t ≠ 0, for the chosen constants. The proof invokes the maximum principle and reduces the boundary check to the roots of a degree-13 polynomial, whose seven real roots are listed after a computer-assisted 'multiplying out'. This inequality is what lets the author replace the infinite zero sum by logarithmic derivatives of ξ_K at the point 1/2 + d + iT, which are then estimated by adding the cont

What would settle it

Compute H(b,t) from Lemma 2.1 on a fine grid over |b| ≤ 1/2 and 0 < t ≤ 6 using interval arithmetic; if any sample point gives H > 0, the lemma is false. Equivalently, verify by independent high-precision computation that the degree-13 polynomial h′(t) has exactly seven real roots and that h takes negative values at the three local maxima; a fourth real root or a positive local maximum would refute the proof.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for every number field K with degree n_K and discriminant d_K, and for every T ≥ 1, |N_K(T) − (T/π) log(d_K (T/(2πe))^{n_K}) − 1.919| ≤ 0.194(log d_K + n_K log T) + 5.543 n_K + 0.462. Corollary 1.2 specializes to the Riemann zeta function: |N(T) − (T/(2π)) log(T/(2πe))| ≤ 0.097 log T + 3.962 for T ≥ 1, which improves on the best previously published explicit bound. The proof starts from the argument principle for ξ_K, writes the logarithmic derivative as an absolutely convergent sum over nontrivial zeros via the Weierstrass factorization, and groups zeros with their functional-equation partners. The contribution of each pair is controlled by the inequality f

Load-bearing premise

The load-bearing premise is the claim, verified only by an unverified computer calculation, that a certain explicit two-variable function is non-positive on its whole domain. If that claim is wrong, the main theorem's error bound does not follow.

Editorial extensions

If this is right

Theorem 1.1 gives uniform explicit bounds for N_K(T) for every number field K and every T ≥ 1, with constants that improve all previously known explicit results.
Corollary 1.2 yields an explicit error term for the Riemann zeta function with logarithmic coefficient 0.097, less than half of the previous best coefficient.
The paper notes that the method is in principle adaptable to any L-function with an Euler product and functional equation, provided an analogue of Lemma 2.1 can be proved.
The constants in Lemma 2.3 and Lemma 2.7 can be slightly improved by evaluating more terms in the Euler-product sum, so the numerical values are not final within the method.

Reading between the lines

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

Because Lemma 2.1's numerical check is not formally certified, the theorem's validity currently rests on the reliability of that check; supplying an interval-arithmetic proof of H(b,t) ≤ 0 would convert the result into a fully rigorous theorem.
The four parameters in Lemma 2.1 were chosen by strategic testing; a systematic nonlinear optimization could find an admissible set with smaller da1, further shrinking the constant 0.194.
The same harmonic-inequality template could be applied to other L-functions in a general class, but the analogue of Lemma 2.1 would have to be proved separately for each family; this seems to be the main obstacle to transferring the method.
If combined with a verified computational framework, the method could be used to generate explicit zero-free regions or bounding boxes for zeros, with applications to prime-number estimates in number fields.

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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

Colleague,

Read the Amberger paper on counting zeros of Dedekind zetas. The core idea is new: an operator-based decomposition, inspired by Turing's method for S1(T), that lets the author bound the zero-counting error term through a cleverly chosen 'pole' combination. The claimed constants, e.g., 0.194(log d_K + n_K log T)+5.543 n_K+0.462, and the Riemann corollary 0.097 log T+3.962, are indeed improvements over HSW21 and earlier work. If they hold, this is a solid contribution to explicit analytic number theory.

The structure is clean: decompose the completed zeta, estimate the polynomial, discriminant, gamma, and zeta pieces separately. The gamma estimates via Binet are standard. The zeta-piece estimate uses an Euler-product bound and a prime-ideal splitting argument. The author also provides a Mathematica notebook, which is more than we usually get.

The main problem is Lemma 2.1. The whole theorem hinges on showing H(b,t) ≤ 0, and the proof does not do that rigorously. For the vertical sides, the argument reduces to a function h(t) whose derivative's zeros are roots of a degree-13 polynomial; the author states that 'multiplying out' gives seven real roots, and that the local maxima are negative but only by about -0.0002. No Sturm sequence, interval-arithmetic certificate, or explicit T0 is given. That margin is too close to trust floating point. If H is positive anywhere, the bounds on Σf, and hence the main theorem, collapse. Lemma 2.7 has the same issue: 'following a mathematica computation' is used to claim a key inequality (2.14).

One specific reader objection is wrong: the final line equating 0.194 log(d_K (T/(2π))^{n_K}) + 5.899 n_K with 0.194(log d_K + n_K log T) + 5.543 n_K is correct—the log(2π) term accounts for the difference. So the theorem as stated is not internally contradictory on that page. The problem is the unverified numerical lemma, not that arithmetic.

Minor points: Lemma 2.3 asserts 'no local extrema' without proof; probably true but should be checked. The Riemann corollary could compare to modern S(T) bounds but that omission is not serious.

Verdict: the method deserves a serious referee. Send it to review, but the referee should demand a certified verification of the key lemma—interval arithmetic, Sturm sequences, explicit T0—and a similar treatment of Lemma 2.7. With that in hand, the paper would be a genuine improvement. Without it, it's a promising preprint, not a theorem.

Referee Report

2 major / 4 minor

Summary. The paper proposes a new explicit estimate for the number N_K(T) of nontrivial zeros (counted with multiplicity, with boundary weights) of a Dedekind zeta function up to height T. The method follows Turing's idea: after an argument-principle identity, the zero sum is bounded through a two-sided harmonic majorant f(b,t,d) depending on parameters d,a1,a2,a3, and the resulting logarithmic-derivative expressions are estimated factor-by-factor for the completed zeta function. The main theorem claims |N_K(T) - (T/pi) log(d_K (T/(2pi e))^{n_K}) - 1.919| ≤ 0.194(log d_K + n_K log T) + 5.543 n_K + 0.462 for T≥1, and Corollary 1.2 states the analogous Riemann-zeta bound with constants 0.097 and 3.962. The proof is conditional on two substantial numerical assertions: Lemma 2.1, which reduces to a claim about the sign of a function H(b,t) supported by approximate roots of a degree-13 polynomial, and Lemma 2.7, which uses several Mathematica range computations and a first-22-prime summation without certified interval arithmetic.

Significance. If all numerical claims could be made fully rigorous, the announced constants would be a genuine improvement over the comparable explicit bound of Hasanalizade–Shen–Wong (0.228, 23.108, 4.52) and would improve the known Riemann-zeta zero-counting estimate in the same style. The structural idea of replacing the zero-sum by a carefully optimized harmonic majorant is attractive and appears to be adaptable to other L-functions. However, the significance is currently conditional: the two main lemmas rest on non-certified computer computations, and the margins in the key inequality are as small as 10^-4, so the result is not yet established to the standard required for an explicit-number-theory paper. The paper does not ship machine-checked proofs or interval-verified certificates; it only points to an arXiv page for a Mathematica notebook.

major comments (2)

[§2, Lemma 2.1 (Eqs. (2.8)–(2.9))] This lemma is the hinge of the proof: it bounds the zero-summand f(b,t,d) and thereby controls E2 and E3. The proof reduces the boundary check to h(t)=H(1/2,t) and then states that h'(t) has zeros which are the roots of a degree-13 polynomial, that there are seven real roots, and that the local maxima have numerically small negative values h(t2)≈-0.00019, h(t4)≈-0.00022, h(t6)≈-0.00015. No Sturm sequence, interval-arithmetic enclosure, or machine-checked certificate is provided. Because the margins at the local maxima are of order 10^-4, ordinary floating-point evaluation is not a rigorous proof; a rounding error of that size would change the sign and invalidate Lemma 2.1, hence also the bounds on E2,E3 and Theorem 1.1. In addition, the appeal to 'large |t|' via (2.9) leaves T0 unspecified and the O-term unquantified. This is a load-bearing gap and must be repaired by certified computati
[§2, Lemma 2.7 (Eq. (2.14) and Eq. (2.16)) and Lemma 2.6] Lemma 2.7 estimates E_u(ζ_K) through q1 and concludes E_u(ζ_K)≤5.633 n_K. The proof relies on non-certified Mathematica assertions: the statement that max_x q1(α,x)≤max_x f2(α,x) for all α≥4 ('Following a mathematica computation...', Eq. (2.14)), the numerical evaluation of the sum over the first 22 primes (≤1.1084), and the tail estimate (4.5243). The same kind of unverified 'computing the range with mathematica' steps appears in Lemma 2.6 (U_1,1+U_B1, L_1,1+L_B1, etc.). These numerical inputs determine the additive constants in Theorem 1.1; without interval-verified bounds or exact inequalities, the theorem is not established. A link to the arXiv abstract page is not a substitute for an included, verified computation. This issue is fixable in principle, but it is load-bearing and must be addressed.

minor comments (4)

[§2, Lemma 2.1] The asymptotic in (2.9) appears to omit a factor d^2: a direct expansion gives H(b,t)= -π d^2(a1-a2)/(2t^2)+O(t^{-3}), not π/(2t^2)(a2-a1)+O(t^{-3}). The sign is still negative, but T0 should be chosen with the correct coefficient and the O-term should be quantified.
[§2, final combining paragraph] The constants in the combining paragraph do not match Lemmas 2.6 and 2.7: 5.633+0.258=5.891, not 5.899, and the lower-side coefficient is 5.633+0.25=5.883, not 5.891. The displayed inequalities should be corrected and the final arithmetic recomputed. I note that the subsequent equality 0.194 log(d_K(T/(2π))^{n_K}) = 0.194(log d_K+n_K log T)-0.356 n_K is arithmetically correct; the problem lies in the inconsistent intermediate constants, not in that rewrite.
[§2, Lemma 2.7] The text states 'M=p_1000=79'; this should be M=p_22=79. Also, the reference to 'equation (2)' near the end of the proof is dangling, since no equation (2) is numbered.
[General] For reproducibility, the Mathematica notebook should be included as a supplement with version information. Better still, the decisive numerical inequalities should be certified with interval arithmetic or exact root isolation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: constants are internal proof parameters, not fitted predictions; the main risks are omitted numerical rigor in Lemmas 2.1 and 2.7, not circularity.

full rationale

Walking the claimed derivation chain: N_K(T) is expressed via the argument principle and the Hadamard factorization (2.3), then regrouped using the functional equation into N_K(T) = (1/pi) sum f + E_1(xi_K). Lemma 2.1 supplies upper/lower pointwise inequalities for f with constants d, a1, a2, a3, and Lemmas 2.3-2.7 bound the decomposed operators E_u/E_l for s(s-1), d_K^{s/2}, gamma_K, zeta_K. The final theorem is obtained by summing these bounds. The constants (d=0.722, a1=1.07, a2=0.93, a3=0.365) are internal admissibility parameters for Lemma 2.1, found, as the paper states, 'by strategic testing'; they are not fitted to N_K(T) or to the target zero-counting error. No quantity that appears as an output of the theorem is used to select a parameter that is then reported as a prediction. The use of [HSW21] is limited to counting zeros in an absolutely convergent sum (O_K(T log T)) and to contextualizing the previous record; it does not assume the new error bound. There is no self-citation chain or imported uniqueness theorem; [HSW21] is independent external work, and the rest are classical references. The genuinely weak spot is rigor, not circularity: Lemma 2.1's verification of H(b,t) <= 0 relies on 'multiplying out' a thirteenth-degree polynomial with seven real roots and on local maxima values h(t2) approx -0.00019 etc., via a Mathematica computation, with an unspecified T0 for the asymptotic (2.9); Lemma 2.7 similarly says 'Following a mathematica computation' for (2.14). No interval certificate or Sturm/root-isolation proof is supplied. These are omitted-proof/numerical-certification concerns that could invalidate Theorem 1.1 if H became positive, but they do not make the derivation circular. Therefore the circularity score is 0.

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

The proof introduces no new physical or mathematical entities. Its free parameters are the optimized constants d, a1, a2, a3 and a prime cutoff; the main unresolved assumptions are the unverified numerical computations inside Lemmas 2.1 and 2.7.

free parameters (5)

d = 0.722
Shift parameter in the contour and in Lemma 2.1; chosen by 'strategic testing' to minimize da1 under the constraint H ≤ 0.
a1 = 1.07
Weight for the Re(ξ'/ξ) term in Lemma 2.1; chosen to satisfy H ≤ 0 and to make a1 > a2.
a2 = 0.93
Weight for the second-derivative real term; with a1 > a2 it controls H at infinity via the t^{-2} asymptotic.
a3 = 0.365
Weight for the second-derivative imaginary term; used as a 'wild card' to balance the behavior for positive and negative t.
M (prime cutoff) = 79 (p_22)
Truncation in Lemma 2.7 chosen 'to comply with the accuracy tolerance' of Mathematica; the final bound depends on this arbitrary cutoff.

assumptions (6)

standard math ξ_K is entire of order 1 and admits a Weierstrass product; the logarithmic derivative sum over zeros converges
Used in equation (2.3) to represent ξ'_K/ξ_K as Q' + Σ(1/ρ + 1/(s−ρ)); standard complex analysis, citing Conway.
standard math Argument principle counts zeros with multiplicity and the contour deformation/functional equation gives the vertical integral representation
Used in equations (2.1)–(2.2); standard analytic number theory, citing Fischer–Lieb and Neukirch.
standard math Binet's first formula and the stated error bounds for log Γ, ψ, ψ1
Used throughout Lemma 2.6; relies on Watson's cited theorem and analytic continuation of the Gamma function.
standard math The Euler product for ζ_K and the fundamental identity for prime ideal decomposition
Used in Lemma 2.7 to reduce prime-ideal sums to rational-prime sums; standard algebraic number theory, citing Neukirch.
ad hoc to paper Mathematica numerical computations are reliable to the claimed precision, including 'computing the range' and polynomial-root enumerations
Used in Lemmas 2.1, 2.6, and 2.7; no code, interval arithmetic, or certificates are provided, so the numerical conclusions are unverified assumptions.
ad hoc to paper The degree-13 polynomial root locations and max_x q1 inequalities are exact or sufficiently accurate for the proof
Used in Lemma 2.1 ('Multiplying out ... shows ...') and Lemma 2.7 ('Following a mathematica computation shows ...'); these are load-bearing numerical checks.

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Pith. "Pith review of Estimating the number of zeros of Dedekind zeta-functions." pith.science (2026). https://pith.science/paper/NWA32TS7

@misc{pith2026251027444,
  author       = {Pith},
  title        = {Pith review of: Estimating the number of zeros of Dedekind zeta-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWA32TS7}},
  note         = {Machine review of arXiv:2510.27444}
}

abstract

In this article, I derive a new approach to estimate the number of non-trivial zeros of a given Dedekind zeta function with absolute height at most $T\geq1$ counted with multiplicity. The error term in corresponding asymptotic formula improves all previous results, even in the case of the Riemann zeta function.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Counting zeros of Artin $L$-functions
math.NT 2026-06 unverdicted novelty 5.0 of 10

Under Artin's conjecture, an explicit asymptotic counts zeros of Artin L-functions up to height T, giving unconditional formulas for Hecke L-functions.

Reference graph

Works this paper leans on

4 extracted references · cited by 1 Pith paper

[1]

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1984
[1999]

Explicit zero density theorems for Dedekind zeta functions

[KN12] Habiba Kadiri and Nathan Ng. “Explicit zero density theorems for Dedekind zeta functions”. In:Journal of Number Theory132.4 (2012), pp. 748–775. [HSW21] Elchin Hasanalizade, Quanli Shen, and PENG-JIE Wong. “Counting zeros of Dedekind zeta functions”. In:Math. Comput.91 (2021), pp. 277–293. Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, G...

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

¨Uber die Nullstellen der Riemannschen Zetafunktion

[Bac16] R.J. Backlund. “ ¨Uber die Nullstellen der Riemannschen Zetafunktion”. In:Acta Mathematica (1916), pp. 345–375. [Wat20] E.T. Whittaker; G.N. Watson.A course of modern Analysis. 3rd ed. Cambridge University Press,

1916

[2] [1920]

Explicit Bounds for Some Functions of Prime Numbers

[Ros41] Barkley Rosser. “Explicit Bounds for Some Functions of Prime Numbers”. In:American Jour- nal of Mathematics63.1 (1941), pp. 211–232. (Visited on 04/01/2025). [Tur53] A. Turing. “Some calculations of the Riemann zeta-function”. In:Proc. London Math. Soc.3 (1953), pp. 99–117. [Con78] John B. Conway.Functions of one complex variable I. 2nd ed. Spring...

1941

[3] [1978]

Explicit Estimates for the Error Term in the Prime Number Theorem for Arithmetic Progressions

[McC84] K.S. McCurley. “Explicit Estimates for the Error Term in the Prime Number Theorem for Arithmetic Progressions”. In:Mathematics of Computation42.165 (1984), pp. 265–285. [Lie94] W. Fischer; I. Lieb.Funktionentheorie. 5th ed. Vieweg+Taubner Verlag,

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[4] [1999]

Explicit zero density theorems for Dedekind zeta functions

[KN12] Habiba Kadiri and Nathan Ng. “Explicit zero density theorems for Dedekind zeta functions”. In:Journal of Number Theory132.4 (2012), pp. 748–775. [HSW21] Elchin Hasanalizade, Quanli Shen, and PENG-JIE Wong. “Counting zeros of Dedekind zeta functions”. In:Math. Comput.91 (2021), pp. 277–293. Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, G...

2012