I was reading through the MathWorld article on the Reimann Prime Counting Function.
The first step in the definition is clear to me:
$$f(x) = \sum_{p^v < x \text{ and p prime}} \frac{1}{v}$$
Here is the second step:
$$=\sum_{n}\frac{\pi(x^{1/n})}{n}$$
It is not clear to me how the sum of the reciprocal of the power is equal to the sum of the prime counting function divided by all values of $n$.
I would appreciate if someone could show how the second step follows from the first.
Hint:
How many $v$-th powers of primes are there between $1$ and $x$?
Answer: if $p$ is a prime then $p^v<x$ iff $p<x^{1/v}$. Then the number of $v$-th powers of primes between $1$ and $x$ is $\pi(x^{1/v})$.