Define (note this sum is actually finite).
1.) Show .
2.) Given , show .
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
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.