# The probability of relative primality of Gaussian integers

Algorithmic Number Theory

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## 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.

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: We use this factorization to approximate the Gaussian probability to 17 decimal places.

*ζ*_{ G }(2)=*ζ*(2)*L*(2,*χ*), where$$L\left( {2,\chi } \right) = 1 - \frac{1}{{3^2 }} + \frac{1}{{5^2 }} - \frac{1}{{7^2 }} + \cdots .$$

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## References

- [1]Raymond Ayoub.
*An Introduction to the Analytic Theory of Numbers*. American Mathematical Society, Providence, Rhode Island, 1963.Google Scholar - [2]Z. I. Borevich and I. R. Shafarevich.
*Number Theory*. Academic Press, Inc., Orlando, Florida, 1966.Google Scholar - [3]D. E. Knuth.
*The Art of Computer Programming, Vol. 2: Seminumerical Algorithms*. Addison-Wesley, second edition edition, 1981.Google Scholar - [4]Serge Lang.
*Algebraic Number Theory*. Addison-Wesley, Reading, Massachusetts, 1970.Google Scholar - [5]Jack Schwartz. Algebraic number theory. 1961. Notes by Hannah Rosenbaum and Carole Sirovich.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1989