The probability of relative primality of Gaussian integers
 George E. Collins,
 Jeremy R. Johnson
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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 Lseries. For the Gaussian integers we get: ζ _{ G }(2)=ζ(2)L(2,χ), where $$L\left( {2,\chi } \right) = 1  \frac{1}{{3^2 }} + \frac{1}{{5^2 }}  \frac{1}{{7^2 }} + \cdots .$$ We use this factorization to approximate the Gaussian probability to 17 decimal places.
 Raymond Ayoub. An Introduction to the Analytic Theory of Numbers. American Mathematical Society, Providence, Rhode Island, 1963.
 Z. I. Borevich and I. R. Shafarevich. Number Theory. Academic Press, Inc., Orlando, Florida, 1966.
 D. E. Knuth. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms. AddisonWesley, second edition edition, 1981.
 Serge Lang. Algebraic Number Theory. AddisonWesley, Reading, Massachusetts, 1970.
 Jack Schwartz. Algebraic number theory. 1961. Notes by Hannah Rosenbaum and Carole Sirovich.
 Title
 The probability of relative primality of Gaussian integers
 Book Title
 Symbolic and Algebraic Computation
 Book Subtitle
 International Symposium ISSAC '88 Rome, Italy, July 4–8, 1988 Proceedings
 Pages
 pp 252258
 Copyright
 1989
 DOI
 10.1007/3540510842_23
 Print ISBN
 9783540510840
 Online ISBN
 9783540461531
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 358
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
 Additional Links
 Topics
 eBook Packages
 Editors
 Authors

 George E. Collins ^{(1)}
 Jeremy R. Johnson ^{(1)}
 Author Affiliations

 1. Dept. of Computer and Information Sciences, Ohio State University, 43210, Columbus, OH
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