$\log$ transform of the fundamental theorem of arithmetic?

Question

Taking the canonical form of the fundamental theorem of arithmetic in the form: $$n=\prod_{j=1}^\infty p^{m_j}_j \qquad ;m_j\in \Bbb N_0$$ Does anybody know about a $\log n$ transform of this?

Note: also reference to other works are welcome

Answer 1

You don't clarify what "$\log$ transform" of FTA would look like. A trivial reformulation of FTA is that the log of any positive natural can be written uniquely as a $\bf N$-linear combination of logs of primes, but there is little aesthetic appeal in this formulation. Another algebraic version is this:

$$\log\left({\bf Q}^\times_{>0}\right)=\bigoplus_p (\log p){\bf Z}.$$

One application of this log perspective though is in exhibiting an infinite $\bf Q$-linearly independent set of real numbers, thereby proving that $\bf R$ is an infinite-dimensional $\bf Q$-vector space via arithmetic.