3.1.2, due October 4

I'm not sure if I had such a hard time understanding because I just finished taking the test when I read this section or if it was really just that difficult for me to understand. I get the concept that if 1/p diverges, there must be an infinite number of primes, that's simple, and I even understand lemma 3.1.2.1. However, when they introduce the N is where I start getting confused. The Riemann zeta function is conveniently defined so that we can easily just look at the exponent of p and see whether or not the series diverges. I can see why it is nice to have this function defined in such a way, but I don't exactly get their manipulations of it afterward.