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Proceedings of the American Mathematical Society

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The least prime ideal in the Chebotarev density theorem


Authors: Habiba Kadiri, Nathan Ng and Peng-Jie Wong
Journal: Proc. Amer. Math. Soc. 147 (2019), 2289-2303
MSC (2010): Primary 11R44; Secondary 11R42, 11M41, 11Y35
DOI: https://doi.org/10.1090/proc/14384
Published electronically: February 14, 2019
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Abstract: In this article, we prove a new bound for the least prime ideal in the Chebotarev density theorem, which improves the main theorem of Zaman [Funct. Approx. Comment. Math. 57 (2017), no. 1, 115-142] by a factor of $ 5/2$. Our main improvement comes from a new version of Turán's power sum method. The key new idea is to use Harnack's inequality for harmonic functions to derive a superior lower bound for the generalised Fejér kernel.


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Additional Information

Habiba Kadiri
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 3M4 Canada
Email: habiba.kadiri@uleth.ca

Nathan Ng
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 3M4 Canada
Email: nathan.ng@uleth.ca

Peng-Jie Wong
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 3M4 Canada
Email: pengjie.wong@uleth.ca

DOI: https://doi.org/10.1090/proc/14384
Keywords: Chebotarev density theorem, the least prime
Received by editor(s): July 14, 2018
Received by editor(s) in revised form: September 5, 2018
Published electronically: February 14, 2019
Additional Notes: This research was partially supported by the NSERC Discovery grants of the first and second author.
The third author was supported by a PIMS postdoctoral fellowship and the University of Lethbridge.
Communicated by: Amanda Folsom
Article copyright: © Copyright 2019 Habiba Kadiri, Nathan Ng, and Peng-Jie Wong