90m:11165
Collins, George E.(1-OHS-C); Johnson, Jeremy R.(1-OHS-C)
The probability of relative primality of Gaussian integers.
Symbolic and algebraic computation (Rome, 1988), 252--258,
Lecture Notes in Comput. Sci., 358,
Springer, Berlin, 1989.
11R27 (11Y40)
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It is well known that two integers are coprime with probability
$1/\zeta(2)=6/\pi^2$. Here this result is generalized to Gaussian integers,
the probability of coprimality being $1/\zeta_G(2)$, where
$\zeta_G(s)=\zeta(s)L(s,\chi)$, with $\chi$ standing for the primitive
character (mod $4$), is the zeta-function associated with Gaussian integers.
Numerically, $1/\zeta_G(2)=0.663700\cdots$. A similar result holds for the
probability that two ideals in an algebraic number field are coprime.
--Edwin Clark