The least prime ideal in the Chebotarev density theorem

Kadiri, Habiba; Ng, Nathan; Wong, Peng-Jie

doi:10.1090/proc/14384

The least prime ideal in the Chebotarev density theorem
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by Habiba Kadiri, Nathan Ng and Peng-Jie Wong PDF

Proc. Amer. Math. Soc. 147 (2019), 2289-2303

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.

References

J.-H. Ahn and S.-H. Kwon, An explicit upper bound for the least prime ideal in the Chebotarev density theorem, preprint, arxiv.org/abs/1807.00508.
Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Math. Comp. 65 (1996), no. 216, 1717–1735. MR 1355006, DOI 10.1090/S0025-5718-96-00763-6
Max Deuring, Über den Tschebotareffschen Dichtigkeitssatz, Math. Ann. 110 (1935), no. 1, 414–415 (German). MR 1512947, DOI 10.1007/BF01448036
D. K. Dimitrov, Extremal Positive Trigonometric Polynomials, Approximation Theory: A volume dedicated to Blagovest Sendov, (B. Bojanov ed.), pp. 136-157, Darba, Sofia, 2002.
D. R. Heath-Brown, Zero-free regions for Dirichlet $L$-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), no. 2, 265–338. MR 1143227, DOI 10.1112/plms/s3-64.2.265
Habiba Kadiri, Explicit zero-free regions for Dedekind zeta functions, Int. J. Number Theory 8 (2012), no. 1, 125–147. MR 2887886, DOI 10.1142/S1793042112500078
Habiba Kadiri and Nathan Ng, Explicit zero density theorems for Dedekind zeta functions, J. Number Theory 132 (2012), no. 4, 748–775. MR 2887617, DOI 10.1016/j.jnt.2011.09.002
Moritz Kassmann, Harnack inequalities: an introduction, Bound. Value Probl. , posted on (2007), Art. ID 81415, 21. MR 2291922, DOI 10.1155/2007/81415
J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko, A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math. 54 (1979), no. 3, 271–296. MR 553223, DOI 10.1007/BF01390234
J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 409–464. MR 0447191
A. M. Odlyzko, Lower bounds for discriminants of number fields. II, Tôhoku Math. J. 29 (1977), no. 2, 209–216. MR 441918, DOI 10.2748/tmj/1178240652
Jesse Thorner and Asif Zaman, An explicit bound for the least prime ideal in the Chebotarev density theorem, Algebra Number Theory 11 (2017), no. 5, 1135–1197. MR 3671433, DOI 10.2140/ant.2017.11.1135
Asif Zaman, Bounding the least prime ideal in the Chebotarev density theorem, Funct. Approx. Comment. Math. 57 (2017), no. 1, 115–142. MR 3704230, DOI 10.7169/facm/1651