# Applications of the GCD metric

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?

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Without the outer square root (which does no harm to the triangle inequality), the expression amounts to a weighted sum of distances measured by $p$-adic valuations: $d(m,n)^2=\frac12\sum_p|v_p(m)-v_p(n)|\log p$. This should explain why it is a metric. – Emil Jeřábek Sep 19 '11 at 16:30

The distance between two (positive) integers $m,n$ in this metric is then $$\sum_p |v_p(m) - v_p(n)|$$ where the sum goes over the primes and $v_p$ is the p-adic valuation.