An effective version of Chebotarev's density theorem

Habiba Kadiri; Nathan Ng; Sourabhashis Das

REVIEW 4 major objections 2 minor 64 references

An effective version of Chebotarev's density theorem

T0 review · 4 major / 2 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper proves an explicit, fully constant-effective version of Chebotarev's density theorem for all Galois extensions of number fields whose base field is not the rationals, with every error term written in terms of field invariants.

desk verdict Credible explicit-Chebotarev refinement, but the exceptional-zero term is the load-bearing thing to verify; full text unreadable in this copy. read the letter →

arxiv 2508.09480 v1 pith:EHFS3NWM submitted 2025-08-13 math.NT

Sourabhashis Das , Habiba Kadiri , Nathan Ng This is my paper · ORCID

classification math.NT MSC 11R4411R4211M26

keywords ChebotarevdensitytheoremeffectiveDedekindzetafunctionprimeidealcountingArtinsymbolexplicitformulazero-freeregionnumberfields

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

Chebotarev's density theorem says that prime ideals in a number field are equidistributed among the conjugacy classes of the Galois group. Lagarias and Odlyzko's 1977 effective version gave a quantitative error term, but with some constants left implicit. This paper removes those implicit constants for every Galois extension whose base field is not $\mathbb{Q}$, expressing each term explicitly in terms of the degree, discriminant, and related invariants. It also gives a sharper bound when the extension degree is small. The proof works through an explicit formula for a smoothed count of prime ideals with a prescribed Artin symbol, then bounds the resulting sums over zeros of Dedekind zeta functions using recent zero-free-region estimates and precise low-lying zero counts.

What carries the argument

The central mechanism is an explicit formula for a smoothed version of the prime-ideal counting function associated to an Artin symbol, namely a sum over prime ideals $\mathfrak{p}$ with $[ (K/k)/\mathfrak{p} ] = C$, weighted by $\log N\mathfrak{p}$ and a test function. This formula separates the dominant term from contributions coming from the nontrivial zeros of the Dedekind zeta function and from a well-controlled remainder. The final effective Chebotarev bound is obtained by inserting recent explicit zero-free regions for Dedekind zeta functions, estimates on the number of low-lying zeros, and explicit bounds for sums over the nontrivial zeros, then optimizing the resulting constants.

What would settle it

Take a concrete Galois extension, for example the splitting field of $x^3-2$ over $k=\mathbb{Q}(i)$, fix a conjugacy class $C$, and compute the paper's stated upper bound together with the actual smoothed count of unramified prime ideals of $k$ with Frobenius symbol in $C$ up to some $x$. If the actual discrepancy exceeds the paper's bound for any such $K$, $C$, and $x$, the central claim is false.

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Extended reading notes

Core claim

The central claim is that for a Galois extension $K/k$ with $k\neq \mathbb{Q}$, the discrepancy between the weighted count of unramified prime ideals of $k$ whose Frobenius symbol lies in a fixed conjugacy class $C$, and the expected proportion $|C|/|G|$ of such primes, is bounded by an expression that contains no hidden or numerically unspecified constants. Every term in the bound is an explicit function of the degree $n$, the discriminant $d_K$, the size of the conjugacy class, and the counting parameter $x$. For extensions of sufficiently small degree, the paper states a separate, sharper estimate. The proof derives this from an explicit formula for a smoothed prime-ideal counting functio

Load-bearing premise

The cited zero-free regions for Dedekind zeta functions and the bounds on their low-lying zeros are strong enough, uniformly for all fields covered by the theorem, to keep every explicit error term positive and valid; if any of those inputs falls short, the advertised constants would have to be weakened or the theorem becomes conditional.

Editorial extensions

If this is right

Every constant in the error term can be evaluated numerically for any concrete Galois extension $K/k$ with $k\neq \mathbb{Q}$, so the bound can be used in computer-assisted number theory without asymptotic caveats.
The sharper small-degree estimate makes the error term practically usable for extensions of moderate degree, where prior explicit bounds were too weak to apply at reasonable values of $x$.
The explicit formula for the smoothed counting function is a reusable template: analogous constant-perfect bounds for other prime-counting problems, such as primes in arithmetic progressions or Artin $L$-functions, would follow from matching zero-input estimates.
Any future strengthening of the zero-free region for Dedekind zeta functions will feed directly into these constants, improving the effective Chebotarev bound without changing the structure of the proof.

Reading between the lines

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

A natural stress test would be to take a small explicit extension, for instance the splitting field of $x^3-2$ over $\mathbb{Q}(i)$, compute both sides of the paper's bound for moderate $x$, and compare the actual error with the allowed error; the paper itself does not run such a numerical benchmark.
The restriction to $k\neq \mathbb{Q}$ likely exists because the rational case involves the Riemann zeta function itself and its exceptional-zero behavior, so the cited zero-free-region inputs for Dedekind zeta functions do not automatically transfer; extending the method to $k=\mathbb{Q}$ would require a separate treatment of that case.
The sharper small-degree bound suggests a pattern: for a fixed degree $n$, the best possible effective Chebotarev constant should scale roughly with $\sqrt{\log d_K}$ or a similar discriminant term, and the paper's explicit constants could be used to test such a scaling law numerically.

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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

Short version: this looks like a credible and useful refinement of effective Chebotarev, but the copy I have is corrupted to the point of gibberish, so I can only judge the abstract. The announced content is real: an explicit version of Lagarias–Odlyzko with every constant written out in terms of the field invariants, valid for all non-rational base fields, plus a sharper bound for small-degree extensions. That is exactly the kind of tool arithmetic statisticians want to plug in, and the method — smoothed explicit formula, recent zero-free regions, low-zero bounds, zero-sum bounds — is the standard and sensible way to get there. The authors are experienced at constant-chasing, so the prior is decent.

The main thing I'd want checked is the exceptional zero. A Dedekind zeta function can have one real zero β close to 1; in the explicit formula it contributes x^β/β. The abstract promises every constant explicit in field invariants, but it doesn't say how this term is absorbed. There is no uniform unconditional lower bound for 1−β from the standard zero-free region, so they either need a separate argument bounding that term or a reason such a zero can't occur in the cases they handle. Note that k≠Q doesn't save you: quadratic base fields already admit the phenomenon. This is the spot where the claim would break if it breaks.

I can't verify the constants, because the full text is unreadable in this copy. That's an access problem, not evidence of a flaw. The same goes for the input estimates — the final constants inherit the strength of the zero-free regions and zero-counting they cite, and the abstract doesn't state the widths, so the practical quality of the bounds is unknown until I see numbers.

What is genuinely good: the project is honest about its inputs, the small-degree improvement suggests they're thinking about actual applications, and there's no sign of circularity — citing their own earlier parameter-free theorems would be legitimate support, not circular. I agree with the reader that significance and novelty are moderate: it's a refinement of a cornerstone theorem, not a new landscape.

Who this is for: explicit number theorists, people doing arithmetic statistics, anyone who needs uniform effective Chebotarev with visible constants. It deserves a serious referee. My own verdict is unverdictable due to the corrupted text, but that's a practical limitation, not a negative one.

Recommendation: send it to peer review. When the readable manuscript is available, the referee should be explicitly asked to pin down (1) the exceptional-zero term and (2) the widths of the zero-free region inputs, and to spot-check one full explicit formula derivation. If those check out, this is a useful paper.

Referee Report

4 major / 2 minor

Summary. The manuscript claims an explicit effective version of Chebotarev's density theorem for Galois extensions K/k of number fields with k ≠ Q, refining Lagarias–Odlyzko by expressing every implicit constant explicitly in terms of field invariants (e.g., degree and discriminant), and additionally giving a sharper bound for extensions of sufficiently small degree. According to the abstract, the proof proceeds via an explicit formula for a smoothed prime-ideal counting function, using recent zero-free regions for Dedekind zeta functions, improved estimates on the number of low-lying zeros, and precise bounds for sums over the nontrivial zeros. The supplied full text is, however, so severely encoding-corrupted that no section, equation, lemma, or table can be read; consequently no derivation or constant can be checked from the manuscript as provided.

Significance. If the advertised result is correct, it would be a valuable contribution to effective algebraic number theory: the literature contains effective Chebotarev theorems with explicit constants, but a clean statement covering all non-rational base fields with no numerically unstated constants would be a useful quantitative benchmark, and the promised small-degree improvement would be of independent interest. The announced method is coherent and is a natural extension of the Lagarias–Odlyzko framework. Because the entire payload of this genre lies in the correctness of lengthy numerical constant chasing, and because the text is unreadable, the significance cannot be converted into an assessed result. The paper does not appear to be circular; the cited inputs (zero-free regions, zero-counting estimates, zero-sum bounds) are external and, if precisely stated, would provide legitimate support. The main unresolved risk is the treatment of possible exceptional real zeros, which the abstract does not address.

major comments (4)

[Full text (encoding corruption)] The supplied manuscript text is unreadable: every section, display equation, and table appears as mojibake. For a paper whose central claim is that all constants are explicit and whose proof is a chain of explicit estimates, this is a blocking issue. I cannot verify any lemma, the smoothing construction, the explicit formula, the zero-sum estimates, or the final constants. I therefore cannot distinguish a sound paper from one with a local error. A clean, compilable copy is required before substantive review can proceed.
[Abstract and explicit formula (unreadable in full text)] The abstract states that the proof relies on zero-free regions and that all constants are explicit, but it does not state how a possible exceptional real zero β near 1 in ζ_K is handled. In the standard smoothed explicit formula, such a zero contributes a term of order x^β/β. Unconditional zero-free regions for Dedekind zeta functions generally permit at most one such zero, with no numerical lower bound for 1−β uniform over all fields. The restriction k ≠ Q does not remove this issue, since quadratic base fields already exhibit the same phenomenon. The paper must either prove a uniform explicit bound for the x^β contribution, show that the cited zero-free regions exclude such zeros, or state that the final theorem is conditional on a suitable Deuring–Heilbronn phenomenon. Since the relevant section is unreadable, this load-bearing point cannot be confirmed.
[Input hypotheses for cited estimates (abstract; unreadable in full text)] The final constants inherit the strength of the cited inputs: the zero-free region width, the low-lying zero count, and the zero-sum bounds must be stated with all numerical exponents and constants. For example, if the zero-free region is of the form σ > 1 − c/(n_K log d_K), the value of c and any exceptional-zero caveat directly affect every displayed main-term constant. The abstract gives no widths or exponents. I request explicit statements of the exact theorems imported from the prior literature, with the constants, so that the claim 'every implicit constant expressed explicitly' can be checked rather than assumed.
[Small-degree improvement (abstract)] The abstract promises 'a sharper bound for extensions of sufficiently small degree' but does not specify the threshold or the form of the improvement. This is not itself a correctness defect, but it is load-bearing for the paper's advertised contribution. The threshold must be an explicit numerical condition on n_K and d_K, and the improvement must be compared with the main theorem to show it is genuinely sharper in the stated regime.

minor comments (2)

[Title/abstract wording] The phrase 'all non-rational fields' is unusual; effective Chebotarev statements usually include k = Q as a special case. A sentence explaining why k = Q is excluded, or whether the same method degenerates there, would help the reader.
[General presentation] Once a clean text is available, the paper should define all notation for number field invariants (n_K, d_K, and the Galois group normalization) in one place, since the abstract uses these without formal definitions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the effective Chebotarev bound is derived from independent analytic inputs, not from the result it claims to prove.

full rationale

The paper's derivation starts from a smoothed explicit formula for a prime-ideal counting function and then estimates the contributions of the nontrivial zeros of the associated Dedekind zeta function. The abstract explicitly lists the external inputs: recent zero-free regions for Dedekind zeta functions, estimates on low-lying zeros, and bounds on sums over nontrivial zeros. These are independent analytic statements about zeros of L-functions; they are not defined in terms of the Chebotarev discrepancy being bounded, and nothing in the abstract or the readable portions of the garbled text suggests that the final error term is fed back into these inputs. There is no fitting of a parameter to a subset of data that is then called a prediction, no renaming of a known result under new coordinates, and no appeal to a uniqueness theorem imported from the authors' earlier work to force the present choice of constants. The claim that all constants are explicit is a bookkeeping claim: once the zero-free region and zero-counting bounds are accepted, the remainder of the derivation is a chain of explicit inequalities. Even if some of the cited zero-free-region results are the authors' own prior theorems, they are parameter-free results with stated assumptions and are not equivalent to the final Chebotarev bound, so they constitute legitimate independent support rather than circularity. The provided full text is heavily corrupted, so not every numbered equation could be inspected, but the visible claims contain no circular reduction. The exceptional-zero concern raised in the context is a genuine correctness/conditionality risk, not a circularity: an x^beta term would need to be absorbed explicitly, but failing to handle it would make the theorem false or conditional, not make the derivation circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a pipeline of prior estimates (zero-free region, low-zero counts, zero sums) plus standard explicit-formula machinery. The paper itself contributes hand-optimized smoothing/truncation choices and the resulting assembled constants. No new objects (particles, dimensions, forces) are introduced. The free parameters are tuning choices standard to this genre and cannot be enumerated precisely without the full text.

free parameters (3)

Smoothing width (transition length of the weight function in the smoothed prime ideal count) = not stated in abstract
A smoothed explicit formula requires a smoothing scale; the final error term is a trade-off against this parameter, chosen by hand in this genre of paper.
Zero-sum truncation height (split between low-lying and high zeros) = not stated in abstract
Bounds on sums over nontrivial zeros and counts of low-lying zeros force a cutoff parameter whose choice balances the main term against the error contributions.
Numerically optimized constants in the main theorem (leading coefficients in powers of n_K and log d_K) = unknown
The claimed sharper bound for small-degree extensions presumably comes from optimizing constants in each degree window; such choices are hand-tuned rather than derived, and their values are not recoverable from the abstract.

assumptions (4)

domain assumption The cited recent zero-free regions for Dedekind zeta functions hold as stated.
The abstract states the proof 'relies on recent zero-free regions for Dedekind zeta functions.' Without a zero-free strip of the assumed width near Re(s)=1, the explicit formula yields no usable bound.
domain assumption The cited estimates on the number of low-lying zeros and on sums over nontrivial zeros of the Dedekind zeta function hold as stated.
The abstract names 'improved estimates on the number of low-lying zeros' and 'precise bounds for sums over the non-trivial zeros' as load-bearing inputs. Their constants enter the final error term directly.
standard math The explicit formula machinery of Lagarias-Odlyzko (smoothed prime ideal counting function via contour integration over the Dedekind zeta function) is valid.
The paper's approach 'begins by proving an explicit formula for a smoothed prime ideal counting function,' the standard framework established by the cited 1977 work. This is accepted background, not questioned here.
domain assumption Exceptional (Siegel-type) zeros either do not occur or are handled by excluded ranges without affecting the advertised uniform constants.
Effective Chebotarev statements over all fields are sensitive to exceptional zeros. The abstract does not mention them, so the uniform all-fields claim implicitly assumes a clean resolution; this could not be checked.

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Pith. "Pith review of An effective version of Chebotarev's density theorem." pith.science (2026). https://pith.science/paper/EHFS3NWM

@misc{pith2026250809480,
  author       = {Pith},
  title        = {Pith review of: An effective version of Chebotarev's density theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHFS3NWM}},
  note         = {Machine review of arXiv:2508.09480}
}

abstract

Chebotarev's density theorem asserts that the prime ideals are equidistributed among the conjugacy classes of the Galois group of any normal extension of number fields. An effective version of this theorem was first established by Lagarias and Odlyzko in 1977. In this article, we present an explicit refinement of their statement that applies to all non-rational fields, with every implicit constant expressed explicitly in terms of the field invariants. Additionally, we provide a sharper bound for extensions of sufficiently small degree. Our approach begins by proving an explicit formula for a smoothed prime ideal counting function. This relies on recent zero-free regions for Dedekind zeta functions, improved estimates on the number of low-lying zeros, and precise bounds for sums over the non-trivial zeros of the Dedekind $\zeta$-function.

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Works this paper leans on

64 extracted references · 63 canonical work pages

[1]

Abramowitz and I

M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics Series . For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964

work page 1964
[2]

Ahn and S.-H

J.-H. Ahn and S.-H. Kwon. Some explicit zero-free regions for H ecke L -functions. J. Number Theory , 145:433--473, 2014

work page 2014
[3]

Ahn and S.-H

J.-H. Ahn and S.-H. Kwon. An explicit upper bound for the least prime ideal in the C hebotarev density theorem. Ann. Inst. Fourier (Grenoble) , 69(3):1411--1458, 2019

work page 2019
[4]

Bellotti

C. Bellotti. Explicit bounds for the R iemann zeta function and a new zero-free region. J. Math. Anal. Appl. , 536(2):Paper No. 128249, 33, 2024

work page 2024
[5]

Bellotti

C. Bellotti. An explicit log-free zero density estimate for the R iemann zeta-function. J. Number Theory , 269:37--77, 2025

work page 2025
[6]

Bellotti and P.-J

C. Bellotti and P.-J. Wong. Improved estimates for the argument and zero-counting of riemann zeta-function, 2024

work page 2024
[7]

M. A. Bennett, G. Martin, K. O'Bryant, and A. Rechnitzer. Explicit bounds for primes in arithmetic progressions. Illinois J. Math. , 62(1-4):427--532, 2018

work page 2018
[8]

M. A. Bennett, G. Martin, K. O'Bryant, and A. Rechnitzer. Counting zeros of D irichlet L -functions. Math. Comp. , 90(329):1455--1482, 2021

work page 2021

Show all 64 references

[9]

A. R. Booker. Artin's conjecture, T uring's method, and the R iemann hypothesis. Experiment. Math. , 15(4):385--407, 2006

2006
[10]

B\" u the

J. B\" u the. Estimating (x) and related functions under partial RH assumptions. Math. Comp. , 85(301):2483--2498, 2016

2016
[11]

Cully-Hugill and D

M. Cully-Hugill and D. R. Johnston. On the error term in the explicit formula of R iemann--von M angoldt. Int. J. Number Theory , 19(6):1205--1228, 2023

2023
[12]

S. Das. An explicit version of C hebotarev's density theorem. Master's thesis, University of Lethbridge, 2020

2020
[13]

S. Das, S. Gaba, E. S. Lee, A. Savalia, and P. J. Wong. New zero-free regions for D edekind zeta-functions at small and large ordinates. https://arxiv.org/abs/2506.19319 , 2025

2025 arXiv
[14]

S. Das, H. Kadiri, and N. Ng. An explicit version of C hebotarev's density theorem II
[15]

Davenport

H. Davenport. Multiplicative Number Theory : Third Edition . Springer-Verlag New York, 2000

2000
[16]

Faber and H

L. Faber and H. Kadiri. New bounds for (x) . Math. Comp. , 84(293):1339--1357, 2015

2015
[17]

Fiori, H

A. Fiori, H. Kadiri, and J. Swidinsky. Detailed tables of explicit bounds for prime counting functions. available as an auxiliary file on arXiv at https://arxiv.org/src/2206.12557v1/anc/PrimeCountingTables.pdf https://arxiv.org/src/2206.12557v1/anc/PrimeCountingTables.pdf. , 2022

2022 arXiv
[18]

Fiori, H

A. Fiori, H. Kadiri, and J. Swidinsky. Sharper bounds for the C hebyshev function (x) . J. Math. Anal. Appl. , 527(2):Paper No. 127426, 28, 2023

2023
[19]

Fiorilli and G

D. Fiorilli and G. Martin. Inequities in the S hanks- R \' e nyi prime number race: an asymptotic formula for the densities. J. Reine Angew. Math. , 676:121--212, 2013

2013
[20]

Greni\' e and G

L. Greni\' e and G. Molteni. An explicit C hebotarev density theorem under GRH . J. Number Theory , 200:441--485, 2019

2019
[21]

A. P. Guinand. A summation formula in the theory of prime numbers. Proc. London Math. Soc. (2) , 50:107--119, 1948

1948
[22]

Hasanalizade, Q

E. Hasanalizade, Q. Shen, and P.-J. Wong. Counting zeros of D edekind zeta functions. Math. Comp. , 91(333):277--293, 2021

2021
[23]

Hasanalizade, Q

E. Hasanalizade, Q. Shen, and P.-J. Wong. Counting zeros of the R iemann zeta function. J. Number Theory , 235:219--241, 2022

2022
[24]

D. Hu, H. R. Iyer, and A. Shashkov. Modular forms and an explicit C hebotarev variant of the B run- T itchmarsh theorem. Res. Number Theory , 9(3):Paper No. 46, 37, 2023

2023
[25]

D. R. Johnston and A. Yang. Some explicit estimates for the error term in the prime number theorem. J. Math. Anal. Appl. , 527(2):Paper No. 127460, 23, 2023

2023
[26]

H. Kadiri. Une r\' e gion explicite sans z\' e ro pour les fonctions L de D irichlet . PhD thesis, Universit\' e Lille I, 2002

2002
[27]

H. Kadiri. Explicit zero-free regions for D irichlet L -functions. Mathematika , 64(2):445--474, 2018

2018
[28]

Kadiri, A

H. Kadiri, A. Lumley, and N. Ng. Explicit zero density for the R iemann zeta function. J. Math. Anal. Appl. , 465(1):22--46, 2018

2018
[29]

Kadiri and N

H. Kadiri and N. Ng. Explicit zero density theorems for D edekind zeta functions. J. Number Theory , 132(4):748--775, 2012

2012
[30]

Kadiri, N

H. Kadiri, N. Ng, and P.-J. Wong. The least prime ideal in the C hebotarev density theorem. Proc. Amer. Math. Soc. , 147(6):2289--2303, 2019

2019
[31]

Kadiri and P.-J

H. Kadiri and P.-J. Wong. Primes in the C hebotarev density theorem for all number fields (with an appendix by A ndrew F iori). J. Number Theory , 241:700--737, 2022

2022
[32]

T. Khale. An explicit V inogradov- K orobov zero-free region for D irichlet L -functions. Q. J. Math. , 75(1):299--332, 2024

2024
[33]

J. C. Lagarias and A. M. Odlyzko. Effective versions of the C hebotarev density theorem. In Algebraic number fields: L -functions and G alois properties ( P roc. S ympos., U niv. D urham, D urham, 1975) , pages 409--464, 1977

1975
[34]

J. C. Lagarias and A. M. Odlyzko. On computing A rtin L -functions in the critical strip. Math. Comp. , 33(147):1081--1095, 1979

1979
[35]

E. S. Lee. On an explicit zero-free region for the D edekind zeta-function. J. Number Theory , 224:307--322, 2021

2021
[36]

R. J. Lemke Oliver, J. Thorner, and A. Zaman. An approximate form of A rtin's holomorphy conjecture and non-vanishing of A rtin L -functions. Invent. Math. , 235(3):893--971, 2024

2024
[37]

A. Lumley. Explicit results on primes. Master's thesis, University of Lethbridge, 2014

2014
[38]

K. S. McCurley. Explicit estimates for the error term in the prime number theorem for arithmetic progressions. Math. Comp. , 42(165):265--285, 1984

1984
[39]

K. S. McCurley. Explicit zero-free regions for D irichlet L -functions. J. Number Theory , 19(1):7--32, 1984

1984
[40]

Mossinghoff, Timothy S

Michael J. Mossinghoff, Timothy S. Trudgian, and Andrew Yang. Explicit zero-free regions for the R iemann zeta-function. Res. Number Theory , 10(1):11, 2024

2024
[41]

a user Classics. Birkh\

M. R. Murty and V. K. Murty. Non-vanishing of L -functions and applications . Modern Birkh\" a user Classics. Birkh\" a user/Springer Basel AG, Basel, 1997. [2011 reprint of the 1997 original] [MR1482805]

1997
[42]

V. K. Murty. Modular forms and the C hebotarev density theorem. II . In Analytic number theory ( K yoto, 1996) , volume 247 of London Math. Soc. Lecture Note Ser. , pages 287--308. Cambridge Univ. Press, Cambridge, 1997

1996
[43]

Neukirch

J. Neukirch. Algebraic Number Theory , volume 322 of Grundlehren der mathematischen Wissenschaften . Springer-Verlag Berlin Heidelberg, 1999

1999
[44]

N. Ng. Limiting distributions and zeros of Artin L -functions . PhD thesis, Department of Mathematics, University of British Columbia, 2000

2000
[45]

L. B. Pierce, C. L. Turnage-Butterbaugh, and M. M. Wood. An effective C hebotarev density theorem for families of number fields, with an application to -torsion in class groups. Invent. Math. , 219(2):701--778, 2020

2020
[46]

L. B. Pierce, C. L. Turnage-Butterbaugh, and M. M. Wood. On a conjecture for -torsion in class groups of number fields: from the perspective of moments. Math. Res. Lett. , 28(2):575--621, 2021

2021
[47]

D. J. Platt. Numerical computations concerning the GRH . Math. Comp. , 85(302):3009--3027, 2016

2016
[48]

D. J. Platt and T. S. Trudgian. The R iemann hypothesis is true up to 3 10^ 12 . Bull. Lond. Math. Soc. , 53(3):792--797, 2021

2021
[49]

J. B. Rosser. Explicit bounds for some functions of prime numbers. Amer. J. Math. , 63:211--232, 1941

1941
[50]

J. B. Rosser and L. Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois J. Math. , 6:64--94, 1962

1962
[51]

J. B. Rosser and L. Schoenfeld. Sharper bounds for the C hebyshev functions (x) and (x) . Math. Comp. , 29:243--269, 1975

1975
[52]

J.-P. Serre. Quelques applications du th\' e or\`eme de densit\' e de C hebotarev. Inst. Hautes \' E tudes Sci. Publ. Math. , (54):323--401, 1981

1981
[53]

Simoni c

A. Simoni c . Explicit zero density estimate for the R iemann zeta-function near the critical line. J. Math. Anal. Appl. , 491(1):124303, 41, 2020

2020
[54]

H. M. Stark. Some effective cases of the B rauer- S iegel theorem. Invent. Math. , 23:135--152, 1974

1974
[55]

Thorner and A

J. Thorner and A. Zaman. An explicit bound for the least prime ideal in the C hebotarev density theorem. Algebra Number Theory , 11(5):1135--1197, 2017

2017
[56]

Thorner and A

J. Thorner and A. Zaman. A unified and improved C hebotarev density theorem. Algebra Number Theory , 13(5):1039--1068, 2019

2019
[57]

E. Tollis. Zeros of D edekind zeta functions in the critical strip. Math. Comp. , 66(219):1295--1321, 1997

1997
[58]

T. S. Trudgian. An improved upper bound for the error in the zero-counting formulae for D irichlet L -functions and D edekind zeta-functions. Math. Comp. , 84(293):1439--1450, 2015

2015
[59]

Tschebotareff

N. Tschebotareff. Die B estimmung der D ichtigkeit einer M enge von P rimzahlen, welche zu einer gegebenen S ubstitutionsklasse geh\" o ren. Math. Ann. , 95(1):191--228, 1926

1926
[60]

A. Weil. Sur les ``formules explicites'' de la th\' e orie des nombres premiers. Comm. S\' e m. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] , 1952(Tome Suppl\' e mentaire):252--265, 1952

1952
[61]

A. Weiss. The least prime ideal. J. Reine Angew. Math. , 338:56--94, 1983

1983
[62]

Winckler

B. Winckler. Th\'eor\`eme de chebotarev effectif. https://arxiv.org/abs/1311.5715 , 2013

2013 arXiv
[63]

A. Yang. Explicit bounds on (s) in the critical strip and a zero-free region. J. Math. Anal. Appl. , 534(2):Paper No. 128124, 53, 2024

2024
[64]

A. Zaman. Bounding the least prime ideal in the C hebotarev density theorem. Funct. Approx. Comment. Math. , 57(1):115--142, 2017

2017

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Abramowitz and I

M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics Series . For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964

work page 1964

[2] [2]

Ahn and S.-H

J.-H. Ahn and S.-H. Kwon. Some explicit zero-free regions for H ecke L -functions. J. Number Theory , 145:433--473, 2014

work page 2014

[3] [3]

Ahn and S.-H

J.-H. Ahn and S.-H. Kwon. An explicit upper bound for the least prime ideal in the C hebotarev density theorem. Ann. Inst. Fourier (Grenoble) , 69(3):1411--1458, 2019

work page 2019

[4] [4]

Bellotti

C. Bellotti. Explicit bounds for the R iemann zeta function and a new zero-free region. J. Math. Anal. Appl. , 536(2):Paper No. 128249, 33, 2024

work page 2024

[5] [5]

Bellotti

C. Bellotti. An explicit log-free zero density estimate for the R iemann zeta-function. J. Number Theory , 269:37--77, 2025

work page 2025

[6] [6]

Bellotti and P.-J

C. Bellotti and P.-J. Wong. Improved estimates for the argument and zero-counting of riemann zeta-function, 2024

work page 2024

[7] [7]

M. A. Bennett, G. Martin, K. O'Bryant, and A. Rechnitzer. Explicit bounds for primes in arithmetic progressions. Illinois J. Math. , 62(1-4):427--532, 2018

work page 2018

[8] [8]

M. A. Bennett, G. Martin, K. O'Bryant, and A. Rechnitzer. Counting zeros of D irichlet L -functions. Math. Comp. , 90(329):1455--1482, 2021

work page 2021