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Arithmetic functions are defined from natural numbers to complex numbers. Therefore, they are not continuous in the analytic sense and consequently cannot be differentiated analytically. However, we do know that some arithmetic functions are asymptotic to analytic functions. That is, when we connect the points of an arithmetic function on the coordinate plane, it looks as though we could take a derivative, as if it were a curve. Is there a definition in the literature for such a derivative? For example, it is accepted that $\frac{d\pi(n)}{dn} \sim \frac{1}{\log n}$ which is a consequence of the Prime Number Theorem. But how can this derivative $\frac{d\pi(n)}{dn}$ be formally expressed?

$\pi(n)$ function looks like below:

enter image description here

Here you can see it can not be differentiable. But as you scale down the graph, you can see it will look like a curve.

enter image description here

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There is a problem when you try to use asymptotic approximations to define something that is local (as the derivative). Two functions can be asymptotic and still differ by a lot. So suppose that you are trying to define the derivative of an arithmetic function $\alpha$ and you know that $\alpha \sim f$ where $f$ is analytic. It can happen that there is another analytic function $g$ such that $\alpha \sim g$ also, but $f'\ne g'$. So how would you choose which function is the right one to define as the derivative of $\alpha$?

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  • $\begingroup$ Good point. can you give an example for this case? $\endgroup$
    – Severus' Constant
    Commented Jan 19 at 12:51
  • $\begingroup$ Take $\alpha(x)=\pi(x)$, $f(x) = \frac{x}{\log x}$ and $g(x) = \frac{x}{\log x} + \sin x$ $\endgroup$
    – jjagmath
    Commented Jan 19 at 14:42

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