Explicit smoothed prime ideals theorems under GRH
Authors:
Loïc Grenié and Giuseppe Molteni
Journal:
Math. Comp. 85 (2016), 1875-1899
MSC (2010):
Primary 11R42; Secondary 11Y40
DOI:
https://doi.org/10.1090/mcom3039
Published electronically:
October 6, 2015
MathSciNet review:
3471112
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be the Chebyshev function of a number field
. Let
and
. We prove under GRH (Generalized Riemann Hypothesis) explicit inequalities for the differences
and
. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals.
- [1] Eric Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355–380. MR 1023756, https://doi.org/10.1090/S0025-5718-1990-1023756-8
- [2] Eric Bach, Improved approximations for Euler products, Number theory (Halifax, NS, 1994), CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1995, pp. 13-28.
- [3] Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Math. Comp. 65 (1996), no. 216, 1717–1735. MR 1355006, https://doi.org/10.1090/S0025-5718-96-00763-6
- [4] Karim Belabas, Francisco Diaz y Diaz, and Eduardo Friedman, Small generators of the ideal class group, Math. Comp. 77 (2008), no. 262, 1185–1197. MR 2373197, https://doi.org/10.1090/S0025-5718-07-02003-0
- [5] Karim Belabas and Eduardo Friedman, Computing the residue of the Dedekind zeta function, Math. Comp. 84 (2015), no. 291, 357–369. MR 3266965, https://doi.org/10.1090/S0025-5718-2014-02843-3
- [6] Johannes Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, Séminaire de Théorie des Nombres, Paris 1988–1989, Progr. Math., vol. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 27–41. MR 1104698
- [7] Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
- [8] L. Grenié and G. Molteni, Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH, To appear in Math. Comp., DOI: https://doi.org/10.1090/mcom3031.
- [9] L. Grenié and G. Molteni, Zeros of Dedekind zeta functions under GRH, arXiv:1407.1375, To appear in Math. Comp., DOI: https://doi.org/10.1090/mcom/3024.
- [10] 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
- [11] Youness Lamzouri, Xiannan Li, and Kannan Soundararajan, Conditional bounds for the least quadratic non-residue and related problems, Math. Comp. 84 (2015), no. 295, 2391–2412. MR 3356031, https://doi.org/10.1090/S0025-5718-2015-02925-1
- [12] Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723
- [13] A. M. Odlyzko, Some analytic estimates of class numbers and discriminants, Invent. Math. 29 (1975), no. 3, 275–286. MR 376613, https://doi.org/10.1007/BF01389854
- [14] A. M. Odlyzko, Discriminant bounds, http://www.dtc.umn.edu/~odlyzko/unpublished/index.html, 1976.
- [15] A. M. Odlyzko, Lower bounds for discriminants of number fields, Acta Arith. 29 (1976), no. 3, 275–297. MR 401704, https://doi.org/10.4064/aa-29-3-275-297
- [16] A. M. Odlyzko, Lower bounds for discriminants of number fields. II, Tôhoku Math. J. 29 (1977), no. 2, 209–216. MR 441918, https://doi.org/10.2748/tmj/1178240652
- [17] 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
- [18] 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
- [19] J. Oesterlé, Versions effectives du théorème de Chebotarev sous l'hypothèse de Riemann généralisée, Astérisque 61 (1979), 165-167.
- [20] The PARI Group, Bordeaux, megrez number field tables, 2008, Package nftables.tgz from http://pari.math.u-bordeaux.fr/packages.html.
- [21] The PARI Group, Bordeaux, PARI/GP, version 2.6.0, 2013, available from http://pari.math.u-bordeaux.fr/.
- [22] Barkley Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211–232. MR 3018, https://doi.org/10.2307/2371291
- [23] Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
- [24] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, https://doi.org/10.1007/BF01405166
- [25] Timothy S. Trudgian, An improved upper bound for the argument of the Riemann zeta-function on the critical line II, J. Number Theory 134 (2014), 280–292. MR 3111568, https://doi.org/10.1016/j.jnt.2013.07.017
- [26] Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575
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Additional Information
Loïc Grenié
Affiliation:
Dipartimento di Ingegneria gestionale, dell’informazione e della produzione, Università di Bergamo, viale Marconi 5, 24044 Dalmine, Italy
Email:
loic.grenie@gmail.com
Giuseppe Molteni
Affiliation:
Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italy
Email:
giuseppe.molteni1@unimi.it
DOI:
https://doi.org/10.1090/mcom3039
Received by editor(s):
October 18, 2013
Received by editor(s) in revised form:
January 15, 2014, June 6, 2014, and January 14, 2015
Published electronically:
October 6, 2015
Article copyright:
© Copyright 2015
American Mathematical Society