In the pre-MO era, I once realized that on the integers, the function

$$ d(m, n) := \sqrt{\log \frac{\sqrt{mn}} {\text{gcd}(m,n)}}\ , $$

is a metric (all properties are easily verified; in fact this is a Hilbertian metric).

Now that I got reminded of it, I wanted to ask

Has anyone seen this metric before? If so, in what context? What could be potential applications of this metric?

**Added** Given the formulae mentioned by Emil and Quid, an additional thing that I am wondering about is the following (I hope I'm not being obtuse):

Does the fact that $d(m,n)=\|\phi(m)-\phi(n)\|$, where $\phi(m)$ and $\phi(n)$ lie in some Hilbert space, have any interesting ramifications?