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?