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.