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I am not talking here about deriving a constant in the usual sense (which gives $0$ as everyone knows).

I have once heard of a derivative operator with relaxed conditions that was defined over the real numbers whose purpose at first was to study number theory.

Usually, two of the essential properties that we demand on a differential operator is its linearity and its way of acting over products, i.e. if I call my differential operator $D$, :

$$ D(f+g) = Df + Dg, \quad D(fg) = (Df)g + fDg. $$

Let's suppose we delete the linearity condition and keep only the product rule. Define $D(1) = 0$, $D(p) = 1$ if $p$ is prime and suppose $D$ satisfies the product relation over products. It is quite unclear in my memory at which point this operator could do (i.e. on which things could we apply it, was it restricted to algebraic numbers, or could it go over any real, complex?) Between the two lines I'm detailing what I know about this operator.


For instance, $D(4) = 2D(2) + 2(D(2)) = 2(1) + 2(1) = 4$. Hence $D(4) = 4$ is equal to its own derivative. Right now my definition only make sense with integers, since we can factor an integer $n$ in its prime decomposition and then apply the product rule to find the answer. We can also find the derivative of a fraction : $$ 0 = D(1) = D(q/q) = q D(1/q) + 1/q D(q) \quad \Rightarrow \quad D(1/q) = -\frac{D(q)}{q^2}. $$ hence we can deduce the usual formula $D(p/q) = \frac{qD(p) - pD(q)}{q^2}$ by a similar argument. By an inductive argument, we can also show things as $D(a^n) = na^{n-1}$, and define this operator on algebraic numbers (at least for some that I know), for instance if $a$ is positive, $$ D(a) = a^{1/2} D(a^{1/2}) + a^{1/2} D(a^{1/2}) = 2a^{1/2} D(a^{1/2}), \quad \Rightarrow D(a^{1/2}) = \frac{D(a)}{2a^{1/2}} $$ and note here the similarity for the formula for deriving the function $1/f(x)$ in the real differentiable functions system.


If anyone has heard of such an operator over numbers which I have tried to describe as much as possible, can anyone tell me if there are any known results related to number theory that uses this tool to lead to some demonstrations that are useful? Some facts about primes, disivibility, writing a number as a sum of things, I don't know, just tell me what you know. I'd love to hear about it.

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I have heard of this: en.wikipedia.org/wiki/Arithmetic_derivative –  Dan Brumleve May 17 '11 at 4:37
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I have heard of it too, but this page basically defines it and says that "it is related to famous conjectures" such as the hundreds of year old ones, but it never says that this idea has done something yet. I wanted to know if this operator did something in its existence. Thanks for the link though, it'll help the readers of the question. Didn't think about putting it there myself. –  Patrick Da Silva May 17 '11 at 4:43
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up vote 8 down vote accepted

Several references at the OEIS.


Added: I especially like this one.

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Besides relating it to old conjectures, are there any links related to actual "results" in there? 'Cause that's what I'm interested in ; I wanna know if this derivative is just a fancy way of stating new things or did it prove itself useful in any way. –  Patrick Da Silva May 17 '11 at 7:20
    
I like the "Added" part. Seems to be just what I wanted. Thanks! –  Patrick Da Silva May 17 '11 at 7:27
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