8
$\begingroup$

Given two randomly chosen positive rational integers, the probability that the two numbers are coprime is $\frac{6}{\pi^2}$. This is also the probability that a positive integer is squarefree. Are there generalizations of these results for Gaussian integers? Or more generally for the ring of integers in an algebraic number field?

$\endgroup$
  • $\begingroup$ I imagine this has been studied before. The probability ought to be $1/\zeta_K(2)$, where $\zeta_K(s)$ is the Dedekind zeta function of the algebraic number field $K$. $\endgroup$ – Peter Humphries Jul 3 '12 at 14:16
  • 3
    $\begingroup$ One should be careful to specify what model for random integers is used, since there is no canonical probability measure on the integers... $\endgroup$ – Denis Chaperon de Lauzières Jul 3 '12 at 16:08
11
$\begingroup$

There are generalizations, see this mathworld article for some results and references. A detailed exposition for arbitrary number fields is given in this paper by G. Collins and J. Johnson

$\endgroup$
  • 2
    $\begingroup$ This paper deals with the probability that two <i>ideals</i> of norm at most $X$ are coprime, and one needs a small extra argument to deal with principal ideals instead (equidistribution of ideals in ideal classes). $\endgroup$ – Denis Chaperon de Lauzières Jul 3 '12 at 16:07
  • $\begingroup$ I believe the authors comment on the distinction... $\endgroup$ – Igor Rivin Jul 3 '12 at 18:22

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.