How to integrate arithmetic functions?

Question

Is it possible to integrate arithmetic functions such as $\Lambda$ or $\pi$?

For example:

If we define $\Lambda_1$ to be a function that is equal to $\ln(n)$ if $n$ is prime, and to $0$ otherwise, what would

$\int_a^b \Lambda_1(t)~dt$ look like? Could it even be calculated?

That's just a specific example, more generally, what would the integral of an arithmetic function represent (it's not really a curve, so it can't represent the area under a curve for one...), and how would they be calculated?

Answer 1

It would just be a summation.

\begin{equation} F(x)=\int_a^x\Lambda_1\,dt=\sum_{a

This would be a step function and is done when, for example, finding the probability distribution function of a discrete probability density function.