What's the derivative and indefinite integral of $\pi(x)$, which gives the number of primes before $x$? I know there's no answer in terms of elementary functions, but is there an answer in terms of other functions or series?

  • $\begingroup$ I would suggest to add the fact that your function is $\pi(x)$ in the title, since it makes ur title more specific and it might help you get more help. :) $\endgroup$ – Pinocchio Jan 23 '14 at 4:20
  • $\begingroup$ @Pinocchio Good idea :) $\endgroup$ – dfg Jan 23 '14 at 4:29
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    $\begingroup$ If by indefinite integral you mean antiderivative, then $\pi(x)$ has none since it has a jump discontinuity at each prime. Did you have some other meaning in mind? $\endgroup$ – Santiago Canez Jan 23 '14 at 5:07
  • $\begingroup$ @SantiagoCanez I just meant a function that I could use to evaluate the area between two points. It doesn't have to be the antiderivative, if theres another way to define the area function. $\endgroup$ – dfg Jan 23 '14 at 5:08

This is how I would think about it.

You know that $\pi(x)$ is approximately equal to $\dfrac{x}{\ln(x)}$ for sufficiently large $x$ (specifically as $x \to \infty$) http://en.wikipedia.org/wiki/Prime_number_theorem).

Thus, now you have expressed $\pi(x)$ as a continuous function that has a better chance of having derivatives and integrals. Now I would plug that into wolfram to see what the integral and derivatives look like.

Apparently, the derivative is $\dfrac{\ln(x) - 1}{\ln^2(x)}$.

However, the obvious disadvantages of this approach is that, the result is only valid for large values of $x$ , its an approximation and it only gives an asymptotic result.

Maybe if you provide more context or why you need this, a better answer might be possible.

Hope it helps!

  • $\begingroup$ Your welcome! I think the key intuition is to note that a function that is discrete doesn't really become differentiable/integrable until you consider something so large that the sum of the discrete and small things starts to become an integral and add to something that makes sense. $\endgroup$ – Pinocchio Mar 2 '14 at 21:23

The derivative of π(x) would be 0 except when x is a prime, when it wouldn't exist since π(x) takes a step at prime values of x.


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