"Inner product" between prime factorizations Let $x \in \mathbb{Q}^+$. Then $x$ can be expressed uniquely as a product over primes:
$$x = \prod_{p \text{ prime}} p^{\nu_p(x)}$$
where $\nu_p(x)$ is the p-adic valuation of $x$. Let $\nu(x) = \langle \nu_2(x), \nu_3(x), \ldots \rangle \in \mathbb{Z}^{\oplus \mathbb{N}}$. Then $\nu(x) + \nu(y) = \nu(xy)$ and $a \nu(x) = \nu(x^a)$. Thus the positive rationals can be viewed as a module where addition and multiplication of two module elements correspond to multiplication and exponentiation of their corresponding rationals. Suppose we define the following "inner product":
$$\nu(x) \cdot \nu(y) = \sum_{p \text{ prime}} \nu_p(x) \nu_p(y)$$
This gives rise to a notion of orthogonality, angles, lengths, and volumes between positive rationals. My question is this: Do this "inner product" and the aforementioned notions have any significance, from the perspective of number theory? Have they been studied before?
For example: if $x,y \in \mathbb{Z}^+$ then $\nu(x) \cdot \nu(y) =0$ iff $x$ and $y$ are coprime. Furthermore, if $x \in \mathbb{Z}^+$ then $\lVert \nu(x) \rVert_1 = \Omega(x)$ and $\lVert \nu(x) \rVert_0 = \omega(x)$.
 A: Here is a high-brow way to interpret your construction.
Recall that the ideles $\mathbb{A}^\times$ of $\mathbb{Q}$ are defined to be the restricted direct product of $\mathbb{R}^\times$ and all $\mathbb{Q}_p^\times$ with respect to the subgroups $\mathbb{Z}_p^\times$.
There is a natural "diagonal" embedding $\mathbb{Q}^\times \subset \mathbb{A}^\times$. Moreover, we can combine all $p$-adic valuations together to get a homomorphism
$$\mathbb{A}^\times \to \bigoplus_p \mathbb{Z}.$$
Your "inner product" is then just given by the composition
$$\mathbb{Q}^\times \times \mathbb{Q}^\times \subset \mathbb{A}^\times \times \mathbb{A}^\times \to \bigoplus_p \mathbb{Z} \times \bigoplus_p \mathbb{Z} \to \bigoplus_p \mathbb{Z} \to \mathbb{Z},$$
where the penultimate map is component wise multiplication and the last map is the obvious sum.
How does this help? Well it illustrates how your construction can be put into the bigger picture of the more natural ideles, which are certainly a very important part of number theory.
But I don't think your construction has "any significance, from the perspective of number theory". Number theorists tend to work with the more natural diagonal embedding $\mathbb{Q}^\times \subset \mathbb{A}^\times$ into the ideles, as the ideles have more structure (e.g. form a topological group)
A: You can find the same idea from Graham, Knuth, and Patashnik's book Concrete Mathematics on page 115 and some related results.
