1. Consider the probability distribution where you are uniform on an arc of the unit circle |z|=1 in the complex plane having arc length 2*pi/P_n where P_n is the nth prime, P_1=2, P_2=3, P_3=5, etc. For concreteness use the arc whose midpoint is the complex number 1. This probability distribution has the property that the expected value of z^k (where z is a complex number sampled from it) is zero whenever the integer k>1 is a multiple of P_n. 2. Now convolve the probability distributions above based on P_1, P_2, ..., P_N together. The resulting probability distribution has the property that the expected value of z^k (where z is a complex number sampled from it) is zero whenever the integer k>1 is divisible by at least one of the first N primes. However, the expected value of z^1, otherwise known as z, is NOT zero, in fact it is some other real number. (Conceivably for some finite set of miraculous N it actually is zero, though for almost all N we shall see it is positive.) This distribution can also be regarded as a distribution on the angle theta of the sampled-point to the real axis, z=exp(i*theta), for -infinity<theta<infinity. In that case, this distribution has (3/pi)*variance(theta) = sum{n=1,2,...,N} (P_n)^(-2) which remains finite even in the limit N-->infinity. The sum on the right converges to 0.452247... Theta has an even-symmetric probability density which rises then falls (unimodal) and is described by a piecewise polynomial(theta) of degree<=N and a finite number of pieces. 3. The finite limit-variance (and finite all-other-moments) means there exists a limit-distribution in the limit N-->infinity. This distribution has a density function. (Its characteristic function can be written neatly as a certain infinite product, incidentally.) It is NOT the uniform density on the circle. It is unimodal and even-symmetric on the real theta-line. To get the density on the circle, you need to "wrap" the line version modulo 2*pi. This density on the circle shares the property with the plain uniform distribution on the circle that the expected value of z^k is ZERO for each integer k with |k|>1. However, when |k|=1 we (unlike plain uniform) do NOT get zero, we get 0.8281 approximately (and of course when k=0 we get 1). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)