Explicit Bound for the Prime Ideal Theorem in Residue Classes

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10737)


We give explicit numerical estimates for the generalized Chebyshev functions. Explicit results of this kind are useful for estimating the computational complexity of algorithms which generate special primes. Such primes are needed to construct an elliptic curve over a prime field using the complex multiplication method.


  1. 1.
    Atkin, A., Morain, F.: Elliptic curves and primality proving. Technical report Project ICSLA RR-1256, INRIA (1990)Google Scholar
  2. 2.
    Chatzigeorgiou, I.: Bounds on the Lambert function and their application to the outage analysis of user cooperation. IEEE Commun. Lett. 17(8), 1505–1508 (2013)CrossRefGoogle Scholar
  3. 3.
    Dupont, R., Enge, A., Morain, F.: Building curves with arbitrary small MOV degree over finite prime fields. J. Cryptol. 18(2), 79–89 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fryska, T.: An estimate of the order of the Hecke-Landau \(\zeta (s,\chi )\)-functions. Funct. Approx. Comment. Math. 16, 55–62 (1988)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fryska, T.: Some effective estimates for the roots of the Dirichlet L-series, II. Funct. Approx. Comment. Math. 16, 21–36 (1988)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Grześkowiak, M.: An algorithmic construction of finite elliptic curves of order divisible by a large prime. Fund. Inform. 136(4), 331–343 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Grześkowiak, M.: Algorithms for relatively cyclotomic primes. Fund. Inform. 125(2), 161–181 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Grześkowiak, M.: Algorithms for pairing-friendly primes. In: Cao, Z., Zhang, F. (eds.) Pairing 2013. LNCS, vol. 8365, pp. 215–228. Springer, Cham (2014). CrossRefGoogle Scholar
  9. 9.
    Grześkowiak, M.: Explicit zero counting formula for Hecke-Landau zeta-functions. Bull. Aust. Math. Soc. 95(3), 400–411 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ingham, A.E.: The Distribution of Prime Numbers. Cambridge University Press, Cambridge (1932)zbMATHGoogle Scholar
  11. 11.
    Israilov, M.: The Laurent expansion of the Riemann Zeta function. Trudy Mat. Inst. Steklov. 158, 98–104 (1981)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Prachar, K.: Primzahlverteilung, Berlin (1957)Google Scholar
  13. 13.
    Radziejewski, M.: On the distribution of algebraic numbers with prescribed factorization properties. Acta Arith. 116(2), 153–171 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland

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