For show , where is the prime counting function.

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- Jun 21st 2010, 01:57 PMchiph588@Log of zeta
For show , where is the prime counting function.

- Jun 21st 2010, 08:46 PMsimplependulum
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 , - Jun 21st 2010, 09:01 PMchiph588@
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

- Jun 22nd 2010, 10:24 AMChris11
Hey, I was just wondering if I could see your proof of the equation in your tag.

- Jun 22nd 2010, 10:30 AMchiph588@
- Jun 22nd 2010, 04:53 PMChris11
simple pendulum. Sorry.

- Jun 23rd 2010, 03:18 AMsimplependulum
- Jun 23rd 2010, 05:54 PMChris11
Lol. No, I wasn't apologizing to you. Yeah, I mean the derivation of the equation in your signature.

- Jun 23rd 2010, 11:12 PMsimplependulum
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 :

- Jun 24th 2010, 09:37 PMChris11
That's a neat derivation. Anyways, I don't think that I'll be able to find a simpaler one, but I'll try! That's part of what math is about, right--simplicity?