This isn't going to be long. I generally read the chapter. I didn't really understand the chapter. I think what he was calling contour integrals is probably called a surface integral on this side of the pond.
Here's a quotable: "Complex smoothness throughout some region is equivalent to the existence of a power series expansion about any point in the region" (139). Here's another: "Perhaps the most important unsolved mathematical problem today is the Riemann hypothesis, which is concerned with the zeros of this analytically extended zeta function, that is, with the solutions of ζ(z)=0."
Well, those are things I don't really understand, but they seemed significant. :-) Throughout this chapter I kept thinking about something he said in his preface about being discouraged by his publisher from going so technical. I don't mind him going deep. I don't mind not understanding. I'm just pretty sure that if I understood this chapter, I still could have explained it a lot better.