The probability of relative primality of Gaussian integers

  • George E. Collins
  • Jeremy R. Johnson
Algorithmic Number Theory

DOI: 10.1007/3-540-51084-2_23

Part of the Lecture Notes in Computer Science book series (LNCS, volume 358)
Cite this paper as:
Collins G.E., Johnson J.R. (1989) The probability of relative primality of Gaussian integers. In: Gianni P. (eds) Symbolic and Algebraic Computation. ISSAC 1988. Lecture Notes in Computer Science, vol 358. Springer, Berlin, Heidelberg

Abstract

In this paper we generalize, to an arbitrary number field, the theorem which gives the probability that two integers are relatively prime. The probability that two integers are relatively prime is 1/ζ(2), where ζ is the Riemann zeta function and 1/ζ(2)=6/π2. The theorem for an arbitrary number field states that the probability that two ideals are relatively prime is the reciprocal of the zeta function of the number field evaluated at two. In particular, since the Gaussian integers are an unique factorization domain, we get the probability that two Gaussian integers are relatively prime is 1/ζG(2) where ζG is the zeta function associated with the Gaussian integers.

In order to calculate the Gaussian probability, we use a theorem that enables us to factor the zeta function into a product of the Riemann zeta function and a Dirichlet series called an L-series. For the Gaussian integers we get: ζG(2)=ζ(2)L(2,χ), where
$$L\left( {2,\chi } \right) = 1 - \frac{1}{{3^2 }} + \frac{1}{{5^2 }} - \frac{1}{{7^2 }} + \cdots .$$
We use this factorization to approximate the Gaussian probability to 17 decimal places.

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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • George E. Collins
    • 1
  • Jeremy R. Johnson
    • 1
  1. 1.Dept. of Computer and Information SciencesOhio State UniversityColumbus

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