# The probability of relative primality of Gaussian integers

- First Online:

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

- 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/*π*^{2}. 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/*ζ*_{G}(2) where *ζ*_{G} is the zeta function associated with the Gaussian integers.

*ζ*

_{G}(2)=

*ζ*(2)

*L*(2,

*χ*), where

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