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Choosing for every prime number $p$ an arbitrary rational value $\alpha_p$ for its formal derivation $p'$, one gets a derivation on the rationals by extending the definition with the Leibniz rule given by $\left(\pm\prod_{p}p^{\nu_p}\right)'=\pm\sum_{p}\alpha_p\frac{\nu_p}{p} \prod_{p}p^{\nu_p}$ where sums and products are over all primes (and one can even consider rational exponents $\nu_p$ but let us stick to integers for simplicity). We have then $(ab)'=a'b+ab'$ and $(a^n)'=na^{n-1}a'$ but $(a+b)'\not=a'+b'$ in general.

Are there choices of the values $\alpha_p$ giving rise to interesting dynamical systems $x\longrightarrow x'$ on the rationals? (The word "interesting" is of course not well-defined but examples are: only finitely many orbits (in the sense that $a,b$ are considered in the same orbit if $a^{(n)}=b^{(m)}$ for natural integers $n,m$), almost all orbits $a^{(\mathbb N)}$ have infinitely many different elements, almost all orbits $a^{(\mathbb N)}$ are finite, ...)

Natural candidates are perhaps $\alpha_p=1$ for all $p$, $\alpha_p$ a (linear combination of) Dirichlet character(s) with rational values or for example $\alpha_p=p$ (this choice fixes all primes). I guess that in general one cannot say much and the study is messy, except if all $\alpha_p$ except one are zero. A first perhaps non-trivial case is given by $\alpha_p=0$ for all primes $p\geq 5$ but $\alpha_2$ and $\alpha_3$ are both non-zero rationals.

Which choices of values for $\alpha_2,\dots$ give rise to an interesting differential calculus?

More precisely, we want a canonical primitive in $\mathbb Q[l_2,l_3,l_5,l_7,l_{11},\dots]$ for every rational number where $l_p$ is a symbol for the logarithm $\int p^{-1}$ of each prime $p$. Are there such choices and if yes has the map $x\longrightarrow \int x$ an interesting dynamical behaviour?

I guess that this can be extended to general number fields (where one has probably to work with ideals instead of numbers).

Motivation: none except fun and perhaps the existence of a good exercise on the Leibniz rule and the chain rule for undergraduates.

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I seem to recall that there are a few articles that have appeared in recent years in the Journal of Integer Sequences on this topic. Keywords: "number derivative". I don't recall your questions specifically being addressed, but nevertheless you might want to take a look. – Pete L. Clark May 26 '10 at 8:28
There is plenty of motivation to ask such a question: if one comes up with a good enough definition of the number derivative, it should be possible to mimic the proof of the Mason-Stothers theorem over Z to prove the abc conjecture. – Qiaochu Yuan May 26 '10 at 8:30
Thanks, indeed my question seems more or less to be a subset of the paper – Roland Bacher May 26 '10 at 8:36
Yes, Victor Ufnarovski studies this problem for quite a long period. – Wadim Zudilin May 26 '10 at 9:31
This answer by dke to another question discusses the case $\alpha_p=1$… – j.c. May 26 '10 at 12:31

Yes, you get some interesting dynamics out. The derivative with all $\alpha_p=1$ goes by the name "the arithmetic derivative," and there are a few references around (Ufnarovski's being the most complete). The short version of the dynamics story is the following: There are many numbers $n$ (e.g., primes, or twice a twin prime) whose higher order derivatives $n^{(k)}$ are eventually zero. There are an easily-described set of numbers (those of the form $n=p^p$ with $p$ prime) which satisfy $n'=n$, and so $n^{(k)}=n$ for all $n$. Finally, there are many numbers (e.g., non-trivial multiples of $p^p$) with $n^{(k)}\rightarrow\infty$. A fairly major open problem is whether or not there are any other possible orbits (i.e., non-trivial cycles).

A comment on importance: Though it's not clear to me that there's any way these links are genuinely helpful, there are some amusingly sneaky ties relating statements about arithmetic derivatives to statements about other classical number theory problems. Ufnarovski gives links to Goldbach's conjecture and the twin primes conjecture, and some undergraduate research I (and colleague Ben Levitt) supervised extends this to Sophie Germain primes and Cunningham chains.

As to the number fields case, one can certainly still play some analogous games, though it's impossible, even in nice cases, to "extend" the arithmetic derivative. Even if the class group is trivial (e.g., $K=\mathbb{Q}(\sqrt{2})$), one has the obvious problem that one would like to have the prime element $\sqrt{2}$ have derivative one, but this is inconsistent with the product rule (if one wants to maintain that $2'=1$.)

I haven't thought much about other values of $\alpha_p$ (though in many cases, I'd imagine you'd get exactly the same dynamics), but as an aside, let me also reference you to Buium's notion of a derivative (also going by the name arithmetic derivative) which is a little fancier, but currently seems to be of more theoretical significance.

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One can get a handle i.e. give a completed direct sum decomposition of all non-additive derivations for rings such as $\mathbb{Z}[\sqrt{2}]$ by separating out the unit group first - see Theorem 4 of…. Of course, non-UFDs will still cause problems. – dke May 26 '10 at 15:21
Interesting. Thanks for the reference. – Cam McLeman May 27 '10 at 16:32

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