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Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH


Authors: Loïc Grenié and Giuseppe Molteni
Journal: Math. Comp. 85 (2016), 889-906
MSC (2010): Primary 11R42; Secondary 11Y40
DOI: https://doi.org/10.1090/mcom3031
Published electronically: October 7, 2015
MathSciNet review: 3434887
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \psi _{\mathbb{K}}$ be the Chebyshev function of a number field $ \mathbb{K}$. Under the Generalized Riemann Hypothesis we prove an explicit upper bound for $ \vert\psi _{\mathbb{K}}(x)-x\vert$ in terms of the degree and the discriminant of $ \mathbb{K}$. The new bound improves significantly on previous known results.


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  • [1] D. A. Goldston, On a result of Littlewood concerning prime numbers, Acta Arith. 40 (1981/82), no. 3, 263-271. MR 664616 (83m:10076)
  • [2] L. Grenié and G. Molteni, Explicit smoothed prime ideals theorems under GRH, to appear in Math. Comp,. 2015, DOI: https://doi.org/10.1090/mcom3039
  • [3] A. E. Ingham, The Distribution of Prime Numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; with a foreword by R. C. Vaughan. MR 1074573 (91f:11064)
  • [4] Habiba Kadiri and Nathan Ng, Explicit zero density theorems for Dedekind zeta functions, J. Number Theory 132 (2012), no. 4, 748-775. MR 2887617, https://doi.org/10.1016/j.jnt.2011.09.002
  • [5] 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 (56 #5506)
  • [6] Serge Lang, Algebraic Number Theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723 (95f:11085)
  • [7] J. E. Littlewood, Two notes on the Riemann zeta-function, Cambr. Phil. Soc. Proc. 22 (1924), 234-242.
  • [8] A. M. Odlyzko, Some analytic estimates of class numbers and discriminants, Invent. Math. 29 (1975), no. 3, 275-286. MR 0376613 (51 #12788)
  • [9] A. M. Odlyzko, Discriminant bounds, http://www.dtc.umn.edu/~odlyzko/unpublished/index.html, 1976.
  • [10] A. M. Odlyzko, Lower bounds for discriminants of number fields, Acta Arith. 29 (1976), no. 3, 275-297. MR 0401704 (53 #5531)
  • [11] A. M. Odlyzko, Lower bounds for discriminants of number fields. II, Tôhoku Math. J. 29 (1977), no. 2, 209-216. MR 0441918 (56 #309)
  • [12] A. M. Odlyzko, On conductors and discriminants, Algebraic number fields: $ L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 377-407. MR 0453701 (56 #11961)
  • [13] A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 1, 119-141 (English, with French summary). MR 1061762 (91i:11154)
  • [14] J. Oesterlé, Versions effectives du théorème de Chebotarev sous l'hypothèse de Riemann généralisée, Astérisque 61 (1979), 165-167.
  • [15] The PARI Group, Bordeaux, Megrez number field tables, 2008, package nftables.tgz from http://pari.math.u-bordeaux.fr/packages.html.
  • [16] The PARI Group, Bordeaux, PARI/GP, version 2.6.0, 2013, available from http://pari.math.u-bordeaux.fr/.
  • [17] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94. MR 0137689 (25 #1139)
  • [18] J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $ \theta (x)$ and $ \psi (x)$, Math. Comp. 29 (1975), 243-269. MR 0457373 (56 #15581a)
  • [19] Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $ \theta (x)$ and $ \psi (x)$. II, Math. Comp. 30 (1976), no. 134, 337-360. MR 0457374 (56 #15581b)
  • [20] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135-152. MR 0342472 (49 #7218)
  • [21] T. S. Trudgian, An improved upper bound for the error in the zero-counting formulae for Dirichlet $ L$-functions and Dedekind zeta-functions, Math. Comp. 84 (2015), no. 293, 1439-1450. MR 3315515, https://doi.org/10.1090/S0025-5718-2014-02898-6
  • [22] B. Winckler, Théorème de Chebotarev effectif, arxiv:1311.5715, http://arxiv.org/pdf/1311.5715v1.pdf, 2013.

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Additional Information

Loïc Grenié
Affiliation: Dipartimento di Ingegneria gestionale, dell’informazione e della produzione, Università di Bergamo, viale Marconi 5, I-24044 Dalmine, Italy
Email: loic.grenie@gmail.com

Giuseppe Molteni
Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133 Milano, Italy
Email: giuseppe.molteni1@unimi.it

DOI: https://doi.org/10.1090/mcom3031
Received by editor(s): December 16, 2013
Received by editor(s) in revised form: April 28, 2014
Published electronically: October 7, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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