Define (note this sum is actually finite).

1.) Show .

2.) Given , show .

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- Jul 7th 2010, 09:50 AMchiph588@Riemann's Prime Counting Function
Define (note this sum is actually finite).

1.) Show .

2.) Given , show . - Jul 7th 2010, 10:51 AMPaulRS
**1.**We have:

Here note that goes with if and only if . Hence we have: , since for n>1, and equal to 1 for n = 1. - see the more general idea here

**2.**

Here we have (*) :

the lower limit is 2 since for u < 2.

Let , then - again the lower limit will be 2 because of (*).

Reverse the summation and the integral (since this works) :

And the rest follows from this post.

In fact we have : whenever we can re-arrange like that. - e.g. setting you see the multiplication of Dirichlet Series.