Is there any use of metric spaces in number theory?

Question

Consider the metric on the natural numbers $d(a,b)=\log(ab/\gcd(a,b)^2)$ and $D(a,b)=2d(a,b)/(\log(a)+\log(b)+d(a,b))$, which is the Steinhaus transform $p=1$ of $d$. This metric has interesting properties, for example $(a^n)_n$ builds a Cauchy sequence. My question is if there are uses of metric spaces in number theory? Thanks for your help! (With this question I do not mean the $p$-adic numbers or real numbers, but something along the lines of the Furstenberg theorem where there is an explicit use of a defined metric over the natural numbers.)