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 ! ) :
where
Therefore ,
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.
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Here's a shorter way to show this but it's very similar to what you did, working backwards though.
.
Since as , we get
I guess you will find another solution but much simpler .
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 :
but
Therefore ,
so we have :