Algebra & Number Theory

A unified and improved Chebotarev density theorem

Jesse Thorner and Asif Zaman

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We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau–Siegel zero is present. Our main theorem also interpolates the strongest unconditional upper bound for the least prime ideal with a given Artin symbol as well as the Chebotarev analogue of the Brun–Titchmarsh theorem proved by the authors.

Article information

Algebra Number Theory, Volume 13, Number 5 (2019), 1039-1068.

Received: 22 March 2018
Revised: 29 November 2018
Accepted: 30 January 2019
First available in Project Euclid: 17 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R44: Distribution of prime ideals [See also 11N05]

distribution of primes Chebotarev density theorem effective uniform binary quadratic forms


Thorner, Jesse; Zaman, Asif. A unified and improved Chebotarev density theorem. Algebra Number Theory 13 (2019), no. 5, 1039--1068. doi:10.2140/ant.2019.13.1039.

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  • E. Bombieri, “Le grand crible dans la théorie analytique des nombres”, Astérisque 18 (1987), 103.
  • J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs 13, Academic Press, London, 1978.
  • D. A. Cox, Primes of the form $x^2 + ny^2$, Wiley, New York, 1989.
  • E. Fouvry and H. Iwaniec, “Gaussian primes”, Acta Arith. 79:3 (1997), 249–287.
  • J. Friedlander and H. Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications 57, American Mathematical Society, Providence, RI, 2010.
  • H. Heilbronn, “On real simple zeros of Dedekind $\zeta $-functions”, pp. 108–110 in Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972), Univ. Colorado, Boulder, Colo., 1972.
  • A. Hildebrand, “On the number of positive integers $\leq x$ and free of prime factors $>y$”, J. Number Theory 22:3 (1986), 289–307.
  • J. C. Lagarias and A. M. Odlyzko, “Effective versions of the Chebotarev density theorem”, pp. 409–464 in Algebraic number fields: $L$-functions and Galois properties (Durham, UK, 1975), edited by A. Fröhlich, Academic Press, London, 1977.
  • J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko, “A bound for the least prime ideal in the Chebotarev density theorem”, Invent. Math. 54:3 (1979), 271–296.
  • C. Lam, D. Schindler, and S. Xiao, “On prime values of binary quadratic forms with a thin variable”, preprint, 2018.
  • U. V. Linnik, “On the least prime in an arithmetic progression, I: The basic theorem”, Rec. Math. [Mat. Sbornik] N.S. 15(57) (1944), 139–178.
  • V. K. Murty, “Modular forms and the Chebotarev density theorem, II”, pp. 287–308 in Analytic number theory (Kyoto, 1996), edited by Y. Motohashi, London Math. Soc. Lecture Note Ser. 247, Cambridge Univ. Press, 1997.
  • M. R. Murty, V. K. Murty, and N. Saradha, “Modular forms and the Chebotarev density theorem”, Amer. J. Math. 110:2 (1988), 253–281.
  • H. M. Stark, “Some effective cases of the Brauer–Siegel theorem”, Invent. Math. 23 (1974), 135–152.
  • J. Thorner and A. Zaman, “An explicit bound for the least prime ideal in the Chebotarev density theorem”, Algebra Number Theory 11:5 (2017), 1135–1197.
  • J. Thorner and A. Zaman, “A Chebotarev variant of the Brun–Titchmarsh theorem and bounds for the Lang–Trotter conjectures”, Int. Math. Res. Not. 2018:16 (2018), 4991–5027.
  • A. Weiss, “The least prime ideal”, J. Reine Angew. Math. 338 (1983), 56–94.
  • A. A. Zaman, Analytic estimates for the Chebotarev density theorem and their applications, Ph.D. thesis, University of Toronto, 2017,