The beauty of mathematics

I've been playing around with the Riemann zeta function recently just as a math refresher, and I've got a few layman's theories floating around I might like to bounce off someone who actually knows something someday. Particularly to do with how it converges to -1/12 when s=-1 and what that (and similar bizarrely converging functions) actually really means. I actually kind of wonder if our common conception of what an integer really is might be all wrong. One thing that interests me is that between complex numbers and the identity between e and cos/sin, is that it seems like numbers often lend themselves remarkably well to lying on a curve or a circle even when you'd think they wouldn't. Also, the Riemann zeta function is pretty closely related to the harmonic function. What if instead of thinking of integers as discrete objects, we need to think of them as waves, with a frequency and interference?