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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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    $\begingroup$ 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. $\endgroup$
    – Emil Jeřábek
    Commented Sep 19, 2011 at 16:30
  • $\begingroup$ How do you construct the embedding $\phi$? $\endgroup$
    – user6671
    Commented Feb 7, 2020 at 6:26

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The closest thing I am aware of would be a word metric. More precisely the word metric on the group of positive rationals (with multiplication) relative to the natural generating set formed by the prime numbers and their reciprocals.

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.

As per Emil's comment your metric is then closely related to a 'weighted version' of this.

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