# $\log$ transform of the fundamental theorem of arithmetic? [closed]

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

## closed as not a real question by Gottfried Helms, Zev Chonoles, Davide Giraudo, Martin, Grigory MJun 15 '13 at 9:38

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question.

• What do you mean by a "$\log n$ transform" of the theorem? There is an internal direct sum decomposition of additive groups $\sum_{n\ge1}(\log n){\bf Z}=\bigoplus_p (\log p){\bf Z}$, if that's at all relevant. – anon Jun 4 '13 at 20:50
• What about it? What is it that you want to know that you don't already know? – anon Jun 4 '13 at 21:24
• I have never seen somewhere this form, although it seems so trivial. I was just wondering whether someone knows about this form elsewhere. – al-Hwarizmi Jun 4 '13 at 21:37
• There is an answer on MSE somewhere (of Bill D) that proves $\{\log p\}$ are independent over the rationals using FToA. Otherwise I don't see any point in a text mentioning this form. – anon Jun 4 '13 at 22:14
• No, it was this one. – anon Jun 5 '13 at 6:09

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.
• @al-Hwarizmi What latter possibility are you referring to? The direct sum isomorphism in the middle of this answer should be a basic exercise if you know abelian group theory and requires no reference outside of such. Using logs to show $\bf R$ is an infinite-dimensional $\bf Q$-vector space is also a pretty basic exercise, and I already linked to a reference in the comments. – anon Jun 15 '13 at 15:41