I've attached a picture of the problem.

This is a "challenge question" from my complex analysis book. I've seen the Riemann zeta function mentioned here and there, so I thought it could be fun to try to do something with it.

I tried looking it up on wikipedia, but there they just give this infinite sum as the "definition" of the function, which leads me to believe that this does in fact converge uniformly. However, using the limit definition of uniform convergence seems like a nightmare to do.

I'm therefore wondering if there is some obvious shortcut here or something. Any help would be appreciated!