Abstract
Let L/K be a Galois extension of number fields with Galois group G. We show that if the density of prime ideals in K that split totally in L tends to 1/∣G∣ with a power saving error term, then the density of prime ideals in K whose Frobenius is a given conjugacy class C ⊂ G tends to ∣C∣/∣G∣ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros of ζL(s)/ζK(s).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aramata H. Über die Teilbarkeit der Dedekindschen Zetafunktionen. Proc Imp Acad Tokyo, 1933, 9: 31–34
Artin E. Beweis des allgemeinen Reziprozitätsgesetzes. Abh Math Sem Univ Hamburg, 1927, 5: 353–363
Brauer R. On Artin’s L-series with general group characters. Ann of Math (2), 1947, 48: 502–514
Chebotarev N. Détermination de la densité des nombres premiers appartenants à une classe donnée de substitutions, I (in Russian). Bull Acad Sci Russie, 1923, 17: 205–230
Chebotarev N. Détermination de la densité des nombres premiers appartenants à une classe donnée de substitutions, II (in Russian). Bull Acad Sci Russie, 1923, 17: 231–250
Chebotarev N. Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören. Math Ann, 1925, 95: 191–228
Foote R, Ginsberg H, Murty V K. Heilbronn characters. Bull Amer Math Soc, 2015, 52: 465–496
Foote R, Murty V K. Zeros and poles of Artin L-series. Math Proc Cambridge Philos Soc, 1989, 105: 5–11
Heilbronn H. On real zeros of Dedekind ζ-functions. Canadian J Math, 1973, 25: 870–873
Lang S. Algebra, revised 3rd ed. Graduate Texts in Mathematics, vol. 211. New York: Springer-Verlag, 2002
MathOverflow. Is there a von Koch-type theorem for the generalized Riemann hypothesis? https://mathoverflow.net/q/181241
Neukirch J. Algebraische Zahlentheorie. Berlin: Springer-Verlag, 1992
Acknowledgements
The first author was supported by the Rényi Intézet Lendület Automorphic Research Group and NKFIH (National Research, Development and Innovation Office) (Grant No. K 143876). The second author was supported in part by a grant from the National Science Foundation, and a Simons Investigator Award from the Simons Foundation. MathOverflow user Lucia is aware of this paper and agrees that it be published in the current state and authorship. The authors are grateful to Lucia. Finally, the authors thank Emmanuel Kowalski and the anonymous referees for valuable comments.
Funding
Open access funding provided by ELKH Alfréd Rényi Institute of Mathematics.
Author information
Authors and Affiliations
Corresponding author
Additional information
On the 50th Anniversary of Chen’s Theorem and the 100th Anniversary of Chebotarev’s Theorem
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Harcos, G., Soundararajan, K. A supplement to Chebotarev’s density theorem. Sci. China Math. 66, 2749–2753 (2023). https://doi.org/10.1007/s11425-022-2141-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-022-2141-1