For show , where is the prime counting function.
This is a problem related to analytic number theory (I believe ) , but since I know nothing about this topic , what i am going to do is only using techniques of integration ...
The partition is following this rule , the k-th integral has the lower limit and upper limit where denotes the k-th prime in ascending order .
Then it can be written as ( for brevity indeed ! ) :
The only thing you left out is the justification of switching around the terms of the sum:
but this isn't too hard to show.
Here's a shorter way to show this but it's very similar to what you did, working backwards though.
Since as , we get
The equation in my signature is obtained by considering the factorization of which is , familiar for us : ,
Rearrange it :
then make the substitution : and consider what will happen then :
so we have :