Derivative of the Prime Counting Function

Question

Can it be sufficient to say that $$\pi'(x)=\sum_{p=\mathrm{primes}}\delta(x-p)$$

since there is a spike in $\pi(x)$ at every prime and does not grow at any other value of $x$?

Answer 1

if $\{p_n\}_{n\in\mathbb{N}}$ is the sequence of primes, $\pi$ is constant in each interval $(p_n, p_{n+1})$. Therefore, in these intervals, $\pi' \equiv 0$. In the rest of the real line, that is, on each prime $p_n$, $\pi$ is not differentiable since it is not even continuous:

$$ \lim_{x \to p_{n+1}^-} \pi(x) = n \neq n+1 = \lim_{x \to p_{n+1}^+} \pi(x) $$