Arithmetic derivative

One might want to define a "derivative" on [math] \mathbb{N}, \mathbb{Z}, \mathbb{Q} [/math].

A candidate is the Lagarias arithmetic derivative (en.wikipedia.org/wiki/Arithmetic_derivative).

Define [math] d(n): \mathbb{N} \cup \{0\} \to \mathbb{N}\cup \{0\} [/math] by the rules
[math] d(p)=1 [/math] when [math] p [/math] is prime.
[math] d(nm)=d(n)*m+n*d(m) [/math] for all [math] n,m\in \mathbb{N}\cup \{0\}[/math]

Then [math] d(0)=d(0*0)=d(0)*0+0*d(0)=0 [/math], and [math] d(1)=d(1*1)=d(1)*1+1*d(1)=2d(1) [/math] gives [math]d(1)=0[/math]. More results are in the image.

This may be extended to [math] d(n): \mathbb{Z} \to \mathbb{Z} [/math] by setting [math] d(-n) = -d(n) [/math] for all [math] n \geq 1 [/math].

A further extension of the derivative to [math] \mathbb{Q} [/math] is obtained by using a quotient rule [math] d(\frac{p}{q}) = \frac{d(p)*q-p*d(q)}{q^2} [/math].

Nice blog post, user. Tell me more, please

Just look at kahler differentials

could anyone invent a more useless tool and even give it a name?

Where are going with this, user?

I mean yes, there are a dozen operators that fulfill the Leibnitz rule, aka derivations.

bookstore.ams.org/surv-222/
it's happening guise

>the spectrum of the integers is “intrinsically curved”

>>the spectrum of the integers is “intrinsically curved”
He's not wrong.

>[math]\color{#b5bd68}{d(nm)=d(n)*m+n*d(m)}[/math]
have you really thought this through

>have you really thought this through
What's wrong with it?