My problem is this: I have an infinite number of (mostly, see later) random points on a unit circle, and I would like to know analytically what their sum would be (Sigmax, Sigmay).

I know Theta for each point; that is, what the arc length is for each point starting at the right-half of the x-axis.

There is a known relationship between Thetan and Thetan+1. Now, I wanted to use my knowledge of each point's Theta to help me find the sums mentioned earlier. I'm going to assume that this will involve the sum of my Theta's (SigmaTheta).

So, the equation I'm after is Sigmax + i * Sigmay = f(SigmaTheta). What I would like to know is, what the heck is the function f defined as?


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A few relationships:
Sigmax = cos(Theta1) + cos(Theta2) + cos(Theta3) ...
Sigmay = sin(Theta1) + sin(Theta2) + sin(Theta3) ...

A possibly useful equation:
nbi = cos(b * ln(n)) + i * sin(b * ln(n))


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When visualizing this, I picture a unit circle with a bunch of little points on it. Then, when we want the sum, a simple average of the points shows up as yet another point inside the circle somewhere (the sum can be easily derived from this average). It seems simple enough to find a method of getting this average from each point's Theta, but I can't find a way to do it. It might be impossible, or ridiculously and self-defeatingly complicated. Oh well.

I'm going on a trip for a couple of days, so I won't be able to reply until I get back. Hope I've explained it well enough without any serious errors, and sorry for the puns in the title

P.S. If you're curious, yeah, this is related to Riemann