pith. sign in

arxiv: 2606.31441 · v1 · pith:J3MGZVVCnew · submitted 2026-06-30 · 🧮 math.NT

Counting zeros of Artin L-functions

Chiara Bellotti , Peng-Jie Wong This is my paper

Pith reviewed 2026-07-01 04:20 UTC · model grok-4.3

classification 🧮 math.NT
keywords Artin L-functionszero countingArtin's holomorphy conjectureHecke L-functionsasymptotic formulanon-trivial zerosDedekind zeta functionsDirichlet L-functions
0
0 comments X

The pith

Assuming Artin's holomorphy conjecture, an explicit asymptotic counts the non-trivial zeros of Artin L-functions up to any height T.

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

The paper derives an explicit asymptotic formula for the number of non-trivial zeros of an Artin L-function below height T, valid for all T at least 1, but only when the L-function satisfies Artin's holomorphy conjecture. This counts zeros attached to Galois representations and controls prime distributions through the explicit formula. The same result produces unconditional explicit zero counts for all Hecke L-functions over any number field. It also sharpens earlier explicit formulas for Dedekind zeta functions, the Riemann zeta function, and Dirichlet L-functions when T is large.

Core claim

Assuming Artin's holomorphy conjecture, the number of non-trivial zeros of an Artin L-function with imaginary part at most T equals an explicit asymptotic expression that holds for every T greater than or equal to 1.

What carries the argument

The explicit asymptotic formula for the zero-counting function of Artin L-functions.

If this is right

  • Yields an unconditional explicit zero-counting formula for Hecke L-functions over any number field.
  • Improves the explicit zero-counting formulas for Dedekind zeta functions and the Riemann zeta function for sufficiently large T.
  • Improves the explicit zero-counting formula for Dirichlet L-functions for sufficiently large T.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The formula may support more precise effective Chebotarev density theorems for Galois extensions of number fields.
  • Numerical verification of zero counts against the asymptotic could provide evidence toward or against the holomorphy conjecture in concrete cases.
  • The method might extend to other families of L-functions once their holomorphy is established.

Load-bearing premise

Artin L-functions are holomorphic on the entire complex plane except for a possible simple pole at s=1.

What would settle it

A direct computation of the zeros of a specific Artin L-function up to a height T larger than the error term, showing a count that deviates from the predicted asymptotic.

read the original abstract

In this article, assuming Artin's (holomorphy) conjecture, we establish an explicit asymptotic formula for the number of non-trivial zeros, up to any given height $T\geq 1$, of Artin $L$-functions. As a consequence, our result yields an unconditional explicit zero-counting formula for Hecke $L$-functions over any number field. In addition, our result improves the recent work of Amberger on Dedekind and Riemann zeta functions and the previous work of Bennett-Martin-O'Bryant-Rechnitzer on Dirichlet $L$-functions for sufficiently large $T$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. Assuming Artin's holomorphy conjecture, the manuscript derives an explicit asymptotic formula for the number N(T) of non-trivial zeros (up to height T ≥ 1) of Artin L-functions attached to representations of Galois groups of number fields. The main term is (T/2π) log(QT/2π) with explicit lower-order terms and error bounds that improve on prior explicit formulae; the result is unconditional for Hecke L-functions (which are known to be entire) and yields improvements for Dedekind zeta functions, the Riemann zeta function, and Dirichlet L-functions when T is sufficiently large.

Significance. The explicit zero-counting formulae, conditional on a standard conjecture, supply a concrete tool for analytic number theory applications that require precise control of zero locations. The unconditional Hecke case follows immediately from known holomorphy and the functional equation. The claimed refinement of the error term (arising from a more careful contour integration that handles the possible pole at s=1 and the Gamma factors) is a technical but useful improvement over the cited works of Amberger and Bennett–Martin–O'Bryant–Rechnitzer for large T.

minor comments (3)
  1. [Introduction] The introduction should state the precise error term in the main asymptotic (currently only alluded to) so that the improvement over Amberger's result can be compared directly without reading the full proof.
  2. [§1] Notation for the conductor Q and the Artin conductor should be fixed once at the beginning and used consistently; minor inconsistencies appear in the statements of Theorems 1.1 and 1.2.
  3. [§4] A short remark on how the new error term behaves numerically for moderate T (say T=100) would help readers assess practical utility, even if the focus is asymptotic.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of the explicit formulae, the unconditional Hecke case, and the technical improvement in the error term for large T. We are pleased that the referee recommends acceptance.

Circularity Check

0 steps flagged

No significant circularity; derivation is conditional on external conjecture and uses standard contour integration

full rationale

The paper derives an explicit asymptotic for the zero-counting function N(T) for Artin L-functions conditional on Artin's holomorphy conjecture, via the argument principle applied to a rectangular contour (standard technique). The result for Hecke L-functions follows unconditionally from their known entirety. No self-citations are load-bearing for the central claim, no parameters are fitted and relabeled as predictions, and the formula is not obtained by renaming a known result or smuggling an ansatz. The derivation chain is self-contained against the stated external assumption and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on one major open assumption and standard background results in analytic number theory; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Artin's holomorphy conjecture
    Invoked explicitly as the hypothesis under which the asymptotic formula holds.

pith-pipeline@v0.9.1-grok · 5616 in / 1095 out tokens · 43501 ms · 2026-07-01T04:20:33.099970+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 15 canonical work pages · 1 internal anchor

  1. [1]

    and Ng, N

    Kadiri, H. and Ng, N. , TITLE =. J. Number Theory , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.jnt.2011.09.002 , URL =

    work page doi:10.1016/j.jnt.2011.09.002 2012

  2. [2]

    Lagarias, J. C. and Montgomery, H. L. and Odlyzko, A. M. , TITLE =. Invent. Math. , FJOURNAL =. 1979 , NUMBER =. doi:10.1007/BF01390234 , URL =

    work page doi:10.1007/bf01390234 1979

  3. [3]

    Lagarias, J. C. and Odlyzko, A. M. , TITLE =. Algebraic number fields:. 1977 , MRCLASS =

    1977

  4. [4]

    McCurley, K. S. , TITLE =. Math. Comp. , FJOURNAL =. 1984 , NUMBER =. doi:10.2307/2007579 , URL =

    work page doi:10.2307/2007579 1984

  5. [5]

    , TITLE =

    Rademacher, H. , TITLE =. Math. Z. , FJOURNAL =. 1959/1960 , PAGES =. doi:10.1007/BF01162949 , URL =

    work page doi:10.1007/bf01162949 1959

  6. [6]

    , TITLE =

    Rosser, B. , TITLE =. Amer. J. Math. , FJOURNAL =. 1941 , PAGES =. doi:10.2307/2371291 , URL =

    work page doi:10.2307/2371291 1941

  7. [7]

    Trudgian, T. S. , TITLE =. Math. Comp. , FJOURNAL =. 2015 , NUMBER =. doi:10.1090/S0025-5718-2014-02898-6 , URL =

    work page doi:10.1090/s0025-5718-2014-02898-6 2015

  8. [8]

    and Kadiri, H

    Faber, L. and Kadiri, H. , TITLE =. Math. Comp. , FJOURNAL =. 2015 , NUMBER =. doi:10.1090/S0025-5718-2014-02886-X , URL =

    work page doi:10.1090/s0025-5718-2014-02886-x 2015

  9. [9]

    Trudgian, T. S. , title =. 2012 , journal =

    2012

  10. [10]

    Trudgian, T. S. , TITLE =. J. Number Theory , FJOURNAL =. 2014 , PAGES =. doi:10.1016/j.jnt.2013.07.017 , URL =

    work page doi:10.1016/j.jnt.2013.07.017 2014

  11. [11]

    Backlund, R. J. , TITLE =. Acta Math. , FJOURNAL =. 1916 , NUMBER =. doi:10.1007/BF02422950 , URL =

    work page doi:10.1007/bf02422950 1916

  12. [12]

    , TITLE=

    Grossmann, J. , TITLE=

  13. [13]

    Von Mangoldt, H. C. F. , TITLE =. Math. Ann. , FJOURNAL =. 1905 , NUMBER =. doi:10.1007/BF01447494 , URL =

    work page doi:10.1007/bf01447494 1905

  14. [14]

    2022 , author =

    Counting zeros of the Riemann zeta function , journal =. 2022 , author =

    2022

  15. [15]

    2024 , journal=

    An explicit sub-Weyl bound for (1/2 + it) , author=. 2024 , journal=

    2024

  16. [16]

    An explicit van der. Indag. Math. , author =. 2016 , pages =

    2016

  17. [17]

    An improved explicit estimate for (1/2 + it) , volume =. J. Number Theory , author =. 2024 , pages =

    2024

  18. [18]

    Explicit bounds for the Riemann zeta-function on the 1-line , author=. Funct. Approx. Comment. Math. , year=

  19. [19]

    2024 , author =

    Explicit bounds for the Riemann zeta function and a new zero-free region , journal =. 2024 , author =

    2024

  20. [20]

    An explicit upper bound for ( 1 + i t ) , volume =. Indag. Math. , author =. 2022 , pages =

    2022

  21. [21]

    2024 , author =

    Explicit bounds on (s) in the critical strip and a zero-free region , journal =. 2024 , author =

    2024

  22. [22]

    2014 , author =

    An improved upper bound for the argument of the Riemann zeta-function on the critical line II , journal =. 2014 , author =

    2014

  23. [23]

    2024 , note=

    Explicit estimates for the logarithmic derivative and the reciprocal of the Riemann zeta-function , author=. 2024 , note=

    2024

  24. [24]

    Counting zeros of Dedekind zeta functions , author=. Math. Comp. , year=

  25. [25]

    Counting zeros of. Math. Comp. , author =. 2021 , pages =

    2021

  26. [26]

    and Odlyzko, A.M

    Csordas, G. and Odlyzko, A.M. and Smith, W. and Varga, R.S. , journal =. A new Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant . , volume =

  27. [27]

    Dudek , title =

    Adrian W. Dudek , title =. Funct. Approx. Comment. Math. , number =

  28. [28]

    On the error term in the explicit formula of Riemann-von Mangoldt II , author=. Funct. Approx. Comment. Math. , year=

  29. [29]

    Bellotti, Chiara , title =

  30. [30]

    Platt, D. J. and Trudgian, T. S. , title =. Bull. Lond. Math. Soc. , volume =. doi:https://doi.org/10.1112/blms.12460 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms.12460 , year =

    work page doi:10.1112/blms.12460

  31. [31]

    Platt, D. J. , year =. Isolating some non-trivial zeros of zeta , volume =. Math. Comp. , doi =

  32. [32]

    and Hughes, Christopher , year =

    Farmer, David and Gonek, S. and Hughes, Christopher , year =. The maximum size of L -functions , volume =. Journal für die reine und angewandte Mathematik (Crelles Journal) , doi =

  33. [33]

    2015 , author =

    An improved explicit bound on | (1/2+it)| , journal =. 2015 , author =

    2015

  34. [34]

    , title =

    Pintz, J. , title =. Journ\'ees arithm\'etiques de Besan. 1987 , zbl =

    1987

  35. [35]

    2023 , note=

    A new upper bound on the smallest counterexample to the Mertens conjecture , author=. 2023 , note=

    2023

  36. [36]

    2024 , note=

    Explicit Bound of | (1+it )| , author=. 2024 , note=

    2024

  37. [37]

    2022 , author =

    On explicit estimates for S(t) , S_1(t) , and (1/2+it) under the Riemann Hypothesis , journal =. 2022 , author =

    2022

  38. [38]

    Mathematische Annalen , volume=

    Bounding and on the Riemann hypothesis , author=. Mathematische Annalen , volume=. 2013 , publisher=

    2013

  39. [39]

    Milinovich , journal =

    Emanuel Carneiro and Andrés Chirre and Micah B. Milinovich , journal =. Bandlimited approximations and estimates for the Riemann zeta-function , urldate =

  40. [40]

    2025 , note=

    A Note on the Phragmen-Lindelof Theorem , author=. 2025 , note=

    2025

  41. [41]

    Bober and Ghaith A

    Jonathan W. Bober and Ghaith A. Hiary , title =. Experimental Mathematics , volume =. 2018 , publisher =

    2018

  42. [42]

    IEEE Trans

    Johansson, Fredrik , title =. IEEE Trans. Comput. , month =. 2017 , issue_date =

    2017

  43. [43]

    Platt, D. J. , title =

  44. [44]

    Mathematics of Computation , author =

    Isolating some non-trivial zeros of zeta , volume =. Mathematics of Computation , author =. 2017 , pages =

    2017

  45. [45]

    NCI HPC Systems , url=

  46. [46]

    BlueCrystal User Guide , url=

  47. [47]

    , title =

    Martinet, J. , title =. Algebraic Number Fields , editor =. 1977 , publisher =

    1977

  48. [48]

    International Journal of Number Theory , volume =

    Wong, Peng-Jie , title =. International Journal of Number Theory , volume =

  49. [49]

    American Journal of Mathematics , volume=

    Modular forms and the Chebotarev density theorem , author=. American Journal of Mathematics , volume=. 1988 , publisher=

    1988

  50. [50]

    Murty, V. K. , title =. Analytic Number Theory , editor =. 1997 , pages =

    1997

  51. [51]

    Ram Murty and V

    M. Ram Murty and V. Kumar Murty and Peng-Jie Wong , title =. Journal of the Ramanujan Mathematical Society , volume =. 2018 , pages =

    2018

  52. [52]

    Estimating the number of zeros of Dedekind zeta-functions

    Estimating the number of zeros of Dedekind zeta-functions , author=. preprint, arXiv:2510.27444 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  53. [53]

    Algebraic number theory , Url =

    Neukirch, J\"urgen , TITLE =. 1999 , PAGES =. doi:10.1007/978-3-662-03983-0 , URL =

    work page doi:10.1007/978-3-662-03983-0 1999

  54. [54]

    Turing, A. M. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1953 , PAGES =. doi:10.1112/plms/s3-3.1.99 , URL =

    work page doi:10.1112/plms/s3-3.1.99 1953

  55. [55]

    Tunnell, Jerrold , TITLE =. Bull. Amer. Math. Soc. (N.S.) , FJOURNAL =. 1981 , NUMBER =. doi:10.1090/S0273-0979-1981-14936-3 , URL =

    work page doi:10.1090/s0273-0979-1981-14936-3 1981

  56. [56]

    , TITLE =

    Langlands, Robert P. , TITLE =. 1980 , PAGES =

    1980

  57. [57]

    Inventiones mathematicae , volume=

    Serre’s modularity conjecture (I) , author=. Inventiones mathematicae , volume=. 2009 , publisher=

    2009

  58. [58]

    Inventiones mathematicae , volume=

    Serre’s modularity conjecture (II) , author=. Inventiones mathematicae , volume=. 2009 , publisher=

    2009

  59. [59]

    International Mathematics Research Notices , volume=

    Modularity of solvable Artin representations of GO (4)-type , author=. International Mathematics Research Notices , volume=. 2002 , publisher=

    2002

  60. [60]

    Mathematics of Computation , year=

    Improved estimates for the argument and zero-counting function of the Riemann zeta-function , author=. Mathematics of Computation , year=

  61. [61]

    Bulletin of the Australian Mathematical Society , volume=

    Explicit zero-counting theorem for Hecke--Landau zeta-functions , author=. Bulletin of the Australian Mathematical Society , volume=. 2017 , publisher=

    2017

  62. [62]

    Journal of Number Theory , volume=

    On the explicit upper and lower bounds for the number of zeros of the Selberg class , author=. Journal of Number Theory , volume=. 2019 , publisher=

    2019